2.1 Notation and conventions

• ○ 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 $a \leq C b$ and $0< c \leq a/b\leq C$ , where the constants c and C are as above unless explicitly noted to the contrary. Unless otherwise specified, uppercase constants are greater than $1$ , and lowercase constants are smaller than $1$ . In some occasions, it is important to keep track of the dependence on a given parameter $\gamma $ ; in that case, we write $a\lesssim _{\gamma } b$ or $a\approx _{\gamma } b$ to emphasize that the implicit constants in the inequalities depend on $\gamma $ .

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

• ○ Given $E\subset \mathbb {R}^{n+1}$ , we write $\operatorname {\mathrm {diam}}(E)=\sup _{x,y\in E}|x-y|$ to denote its diameter.

• ○ Given an open set $\Omega \subset \mathbb {R}^{n+1}$ , we shall use lowercase 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 $\mathbb {R}^{n+1}\setminus \partial \Omega $ ).

• ○ 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 $ .

• ○ If $\partial \Omega $ is bounded, it is always understood (unless otherwise specified) that all surface balls have radii controlled by the diameter of $\partial \Omega $ , that is, if $\Delta =\Delta (x,r)$ , then $r\lesssim \operatorname {\mathrm {diam}}(\partial \Omega )$ . Note that in this way $\Delta =\partial \Omega $ if $\operatorname {\mathrm {diam}}(\partial \Omega )<r\lesssim \operatorname {\mathrm {diam}}(\partial \Omega )$ .

• ○ For $X \in \mathbb {R}^{n+1}$ , we set $\delta (X):= \operatorname {dist}(X,\partial \Omega )$ .

• ○ We let $\mathcal {H}^n$ denote the n-dimensional Hausdorff measure.

• ○ 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 $X\notin A$ .

• ○ We shall use the letter I (and sometimes J) to denote a closed $(n+1)$ -dimensional Euclidean cube with sides parallel to the coordinate axes, and we let $\ell (I)$ denote the side length of I. We use Q to denote dyadic ‘cubes’ on E or $\partial \Omega $ . The latter exist as a consequence of Lemma 2.8 below.

2.2 Some definitions

Definition 2.2 (Harnack chain condition)

Again following [Reference Jerison and Kenig41], we say that $\Omega $ satisfies the Harnack chain condition if there are uniform constants $C_1,C_2>1$ such that for every pair of points $X, X'\in \Omega $ there is a chain of balls $B_1, B_2, \dots , B_N\subset \Omega $ with $N \leq C_1(2+\log _2^+ \Pi )$ , where

(2.1) $$ \begin{align} \Pi:=\frac{|X-X'|}{\min\{\delta(X), \delta(X')\}} \end{align} $$

such that $X\in B_1$ , $X'\in B_N$ , $B_k\cap B_{k+1}\neq \emptyset $ and for every $1\le k\le N$

(2.2) $$ \begin{align} C_2^{-1} \operatorname{\mathrm{diam}}(B_k) \leq \operatorname{dist}(B_k,\partial\Omega) \leq C_2 \operatorname{\mathrm{diam}}(B_k). \end{align} $$

The chain of balls is called a Harnack chain.

We note that in the context of the previous definition if $\Pi \le 1$ we can trivially form the Harnack chain $B_1=B(X,3\delta (X)/5)$ and $B_2=B(X', 3\delta (X')/5)$ , where equation (2.2) holds with $C_2=3$ . Hence, the Harnack chain condition is nontrivial only when $\Pi> 1$ .

Definition 2.5 (Ahlfors regular)

We say that a closed set $E \subset \mathbb {R}^{n+1}$ is n-dimensional Ahlfors regular (AR for short) if there is some uniform constant $C_1>1$ such that

(2.3) $$ \begin{align} C_1^{-1}\, r^n \leq \mathcal{H}^n(E\cap B(x,r)) \leq C_1\, r^n,\qquad x\in E, \quad 0<r<\operatorname{\mathrm{diam}}(E). \end{align} $$

We next recall the definition of the capacity of a set. Given an open set $D\subset \mathbb {R}^{n+1}$ (where we recall that we always assume that $n\ge 2$ ) and a compact set $K\subset D$ , we define the capacity of K relative to D as

$$\begin{align*}\mathop{\operatorname{Cap}_2}\nolimits(K, D)=\inf\left\{\iint_{D} |\nabla v(X)|^2 dX:\, \, v\in \mathscr{C}^{\infty}_{c}(D),\, v(x)\geq 1 \mbox{ in }K\right\}. \end{align*}$$

Definition 2.7 (CDC)

An open set $\Omega $ is said to satisfy the CDC if there exists a uniform constant $c_1>0$ such that

(2.4) $$ \begin{align} \frac{\mathop{\operatorname{Cap}_2}\nolimits(\overline{B(x,r)}\setminus \Omega, B(x,2r))}{\mathop{\operatorname{Cap}_2}\nolimits(\overline{B(x,r)}, B(x,2r))} \geq c_1 \end{align} $$

for all $x\in \partial \Omega $ and $0<r<\operatorname {\mathrm {diam}}(\partial \Omega )$ .

The CDC is also known as the uniform 2-fatness as studied by Lewis in [Reference Lewis45]. Using [Reference Heinonen, Kilpeläinen and Martio28, Example 2.12], one has that

(2.5) $$ \begin{align} \mathop{\operatorname{Cap}_2}\nolimits(\overline{B(x,r)}, B(x,2r))\approx r^{n-1}, \qquad \mbox{for all} \ x\in\mathbb{R}^{n+1} \ \mbox{and} \ r>0, \end{align} $$

and hence, the CDC is a quantitative version of the Wiener regularity, in particular every $x\in \partial \Omega $ is Wiener regular. It is easy to see that the exterior corkscrew condition implies CDC. Also, it was proved in [Reference Zhao51, Section 3] and [Reference Hofmann, Le, Martell and Nyström30, Lemma 3.27] that a set with Ahlfors regular boundary satisfies the CDC with constant $c_1$ depending only on n and the Ahlfors regular constant.

2.3 Dyadic grids and sawtooths

In this section, we introduce a dyadic grid from [Reference Akman, Hofmann, Martell and Toro1, Lemma 2.13] along the lines of that obtained in [Reference Christ10] but using the dyadic structure from [Reference Hytönen and Kairema39, Reference Hytönen and Kairema40, Reference Hofmann, Mitrea, Mitrea and Morris37]:

Lemma 2.8 (Existence and properties of the ‘dyadic grid’, [Reference Akman, Hofmann, Martell and Toro1, Lemma 2.13])

Let $E\subset \mathbb {R}^{n+1}$ be a closed set. Then there exists a constant $C\ge 1$ depending just on n such that for each $k\in \mathbb {Z}$ there is a collection of Borel sets (called ‘cubes’)

$$ \begin{align*} \mathbb{D}_k:=\big\{Q_j^k\subset E:\ j\in\mathfrak{J}_k\big\}, \end{align*} $$

where $\mathfrak {J}_k$ denotes some (possibly finite) index set depending on k satisfying:

1. (a) $E=\bigcup _{j\in \mathfrak {J}_k}Q_j^k$ for each $k\in \mathbb {Z}$ .

2. (b) If $m\le k$ , then either $Q_j^k \subset Q_i^m$ or $Q_i^m\cap Q_j^k= \emptyset $ .

3. (c) For each $k\in \mathbb {Z}$ , $j\in \mathfrak {J}_k$ and $m<k$ , there is a unique $i\in \mathfrak {J}_m $ such that $Q_j^k\subset Q_i^m$ .

4. (d) For each $k\in \mathbb {Z}$ , $j\in \mathfrak {J}_k$ there is $x_j^k\in E$ such that

   $$\begin{align*}B(x_j^k, C^{-1}2^{-k})\cap E\subset Q_j^k \subset B(x_j^k, C 2^{-k})\cap E. \end{align*}$$

In what follows given $B=B(x,r)$ with $x\in E$ , we will denote $\Delta =\Delta (x,r)=B\cap E$ . A few remarks are in order concerning this lemma. Note that within the same generation (that is, within each $\mathbb {D}_k$ ) the cubes are pairwise disjoint (hence, there are no repetitions). On the other hand, we allow repetitions in the different generations, that is, we could have that $Q\in \mathbb {D}_k$ and $Q'\in \mathbb {D}_{k-1}$ agree. Then, although Q and $Q'$ are the same set, as cubes we understand that they are different. In short, it is then understood that $\mathbb {D}$ is an indexed collection of sets, where repetitions of sets are allowed in the different generations but not within the same generation. With this in mind, we can give a proper definition of the ‘length’ of a cube (this concept has no geometric meaning in this context). For every $Q\in \mathbb {D}_k$ , we set $\ell (Q)=2^{-k}$ , which is called the ‘length’ of Q. Note that the ‘length’ is well defined when considered on $\mathbb {D}$ , but it is not well-defined on the family of sets induced by $\mathbb {D}$ . It is important to observe that the ‘length’ refers to the way the cubes are organized in the dyadic grid. It is clear from $(d)$ that $\operatorname {\mathrm {diam}}(Q)\lesssim \ell (Q)$ . When $E=\partial \Omega $ , with $\Omega $ being a 1-sided NTA domain satisfying the CDC condition, the converse holds, hence $\operatorname {\mathrm {diam}}(Q)\approx \ell (Q)$ ; see [Reference Akman, Hofmann, Martell and Toro1, Remark 2.56]. This means that the ‘length’ is related to the diameter of the cube.

Let us observe that if E is bounded and $k\in {\mathbb {Z}}$ is such that $\operatorname {\mathrm {diam}}(E)<C^{-1}2^{-k}$ , then there cannot be two distinct cubes in $\mathbb {D}_k$ . Thus, $\mathbb {D}_k=\{Q^k\}$ with $Q^k=E$ . Therefore, we are going to ignore those $k\in \mathbb {Z}$ such that $2^{-k}\gtrsim \operatorname {\mathrm {diam}}(E)$ . Hence, we shall denote by $\mathbb {D}(E)$ the collection of all relevant $Q_j^k$ , i.e.,

$$ \begin{align*} \mathbb{D}(E):=\bigcup_k\mathbb{D}_k, \end{align*} $$

where, if $\operatorname {\mathrm {diam}}(E)$ is finite, the union runs over those $k\in \mathbb {Z}$ such that $2^{-k}\lesssim \operatorname {\mathrm {diam}}(E)$ . We write $\Xi =2C^2$ , with C being the constant in Lemma 2.8, which is purely dimensional. For $Q\in \mathbb {D}(E)$ , we will set $k(Q)=k$ if $Q\in \mathbb {D}_k$ . Property $(d)$ implies that for each cube $Q\in \mathbb {D}(E)$ , there exist $x_Q\in E$ and $r_Q$ , with $\Xi ^{-1}\ell (Q)\leq r_Q\leq \ell (Q)$ (indeed $r_Q= (2C)^{-1}\ell (Q)$ ), such that

(2.6) $$ \begin{align} \Delta(x_Q,2r_Q)\subset Q\subset\Delta(x_Q,\Xi r_Q). \end{align} $$

We shall denote these balls and surface balls by

(2.7) $$ \begin{align} B_Q:=B(x_Q,r_Q),\qquad\Delta_Q:=\Delta(x_Q,r_Q), \end{align} $$

(2.8) $$ \begin{align} \widetilde{B}_Q:=B(x_Q,\Xi r_Q),\qquad\widetilde{\Delta}_Q:=\Delta(x_Q,\Xi r_Q), \end{align} $$

and we shall refer to the point $x_Q$ as the ‘center’ of Q.

Let $Q\in \mathbb {D}_k$ , and consider the family of its dyadic children $\{Q'\in \mathbb {D}_{k+1}: Q'\subset Q\}$ . Note that for any two distinct children $Q', Q"$ , one has $|x_{Q'}-x_{Q"}|\ge r_{Q'}=r_{Q"}=r_Q/2$ , otherwise $x_{Q"}\in Q"\cap \Delta _{Q'}\subset Q"\cap Q'$ , contradicting the fact that $Q'$ and $Q"$ are disjoint. Also, $x_{Q'}, x_{Q"}\in Q\subset \Delta (x_Q,\Xi r_Q)$ , hence by the geometric doubling property we have a purely dimensional bound for the number of such $x_{Q'}$ , and hence, the number of dyadic children of a given dyadic cube is uniformly bounded.

We next introduce the ‘discretized Carleson region’ relative to $Q\in \mathbb {D}(E)$ , $\mathbb {D}_{Q}=\{Q'\in \mathbb {D}:Q'\subset Q\}$ . Let $\mathcal {F}=\{Q_i\}\subset \mathbb {D}(E)$ be a family of pairwise 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,

$$\begin{align*}\mathbb{D}_{\mathcal{F}}:=\mathbb{D}\setminus\bigcup_{Q_i\in\mathcal{F}}\mathbb{D}_{Q_i}. \end{align*}$$

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,

$$\begin{align*}\mathbb{D}_{\mathcal{F},Q}:=\mathbb{D}_{Q}\setminus\bigcup_{Q_i\in\mathcal{F}}\mathbb{D}_{Q_i}=\mathbb{D}_{\mathcal{F}}\cap\mathbb{D}_Q. \end{align*}$$

We also allow $\mathcal {F}$ to be the empty set in which case $\mathbb {D}_{\tiny {\emptyset}}=\mathbb {D}(E)$ and $\mathbb {D}_{{\tiny {\emptyset}},Q}=\mathbb {D}_Q$ .

In the sequel, $\Omega \subset \mathbb {R}^{n+1}$ , $n\geq 2$ , will be a 1-sided NTA domain satisfying the CDC. Write $\mathbb {D}=\mathbb {D}(\partial \Omega )$ for the dyadic grid obtained from Lemma 2.8 with $E=\partial \Omega $ . In [Reference Akman, Hofmann, Martell and Toro1, Remark 2.56], it is shown that under the present assumptions one has that $\operatorname {\mathrm {diam}}(\Delta )\approx r_{\Delta }$ for every surface ball $\Delta $ and $\operatorname {\mathrm {diam}}(Q)\approx \ell (Q)$ for every $Q\in \mathbb {D}$ . Given $Q\in \mathbb {D}$ , we define the ‘corkscrew point relative to Q’ as $X_Q:=X_{\Delta _Q}$ . We note that

$$ \begin{align*} \delta(X_Q)\approx\operatorname{dist}(X_Q,Q)\approx\operatorname{\mathrm{diam}}(Q). \end{align*} $$

We also introduce the ‘geometric’ Carleson regions and sawtooths. Given $Q\in \mathbb {D}$ , 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 covering of $\Omega $ with nonoverlapping interiors and satisfy

(2.9) $$ \begin{align} 4\operatorname{\mathrm{diam}}(I)\leq\operatorname{dist}(4I,\partial\Omega)\leq\operatorname{dist}(I,\partial\Omega)\leq 40\operatorname{\mathrm{diam}}(I),\qquad\forall I\in\mathcal{W}, \end{align} $$

and

$$ \begin{align*}\operatorname{\mathrm{diam}}(I_1)\approx\operatorname{\mathrm{diam}}(I_2),\,\text{ whenever }I_1\text{ and }I_2\text{ touch}. \end{align*} $$

Let $X(I)$ denote the center of I, let $\ell (I)$ denote the side length 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 $I,J\in \mathcal {W}$ . 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 $\tau \in (\frac 12,1)$ such that for distinct $I,J\in \mathcal {W}$ , we have that $\tau J\cap I^*= {\emptyset }$ . 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 $0<\lambda <\lambda _0/4$ .

Given $\vartheta \in \mathbb {N}$ , for every cube $Q \in \mathbb {D}$ , we set

(2.10) $$ \begin{align} \mathcal{W}_Q^{\vartheta} :=\left\{I \in \mathcal{W}: 2^{-\vartheta}\ell(Q) \leq \ell(I) \leq 2^{\vartheta}\ell(Q), \text { and } \operatorname{dist}(I, Q) \leq 2^{\vartheta} \ell(Q) \right\}. \end{align} $$

We will choose $\vartheta \ge \vartheta _0$ , with $\vartheta _0$ large enough depending on the constants of the corkscrew condition (cf. Definition 2.1) and in the dyadic cube construction (cf. Lemma 2.8) so that $X_Q \in I$ for some $I \in \mathcal {W}_Q^{\vartheta }$ , and for each dyadic child $Q^j$ of Q, the respective corkscrew points $X_{Q^j} \in I^j$ for some $I^j \in \mathcal {W}_Q^{\vartheta }$ . Moreover, we may always find an $I \in \mathcal {W}_Q^{\vartheta }$ with the slightly more precise property that $\ell (Q)/2 \leq \ell (I) \leq \ell (Q)$ and

$$ \begin{align*} \mathcal{W}_{Q_1}^{\vartheta} \cap \mathcal{W}_{Q_2}^{\vartheta} \neq \emptyset , \quad \text { whenever } 1 \leq \frac{\ell(Q_2)}{\ell(Q_1)} \leq 2, \text { and } \operatorname{dist}(Q_1, Q_2) \leq 1000 \ell(Q_2). \end{align*} $$

For each $I \in \mathcal {W}_Q^{\vartheta }$ , we form a Harnack chain from the center $X(I)$ to the corkscrew point $X_Q$ and call it $H(I)$ . We now let $\mathcal {W}_{Q}^{\vartheta , *}$ denote the collection of all Whitney cubes which meet at least one ball in the Harnack chain $H(I)$ with $I \in \mathcal {W}_Q^{\vartheta }$ , that is,

$$ \begin{align*} \mathcal{W}_{Q}^{\vartheta, *}:=\{J \in \mathcal{W}: \text{ there exists } I \in \mathcal{W}_Q^{\vartheta} \text{ such that } H(I) \cap J \neq \emptyset \}. \end{align*} $$

We also define

$$ \begin{align*} U_{Q}^{\vartheta} :=\bigcup_{I \in \mathcal{W}_{Q}^{\vartheta, *}}(1+\lambda) I=: \bigcup_{I \in \mathcal{W}_{Q}^{\vartheta, *}} I^{*}. \end{align*} $$

By construction, we then have that

$$ \begin{align*} \mathcal{W}_{Q}^{\vartheta} \subset \mathcal{W}_{Q}^{\vartheta, *} \subset \mathcal{W} \quad \text{and}\quad X_Q \in U_Q^{\vartheta}, \quad X_{Q^{j}} \in U_{Q}^{\vartheta}, \end{align*} $$

for each child $Q^j$ of Q. It is also clear that there is a uniform constant $k^*$ (depending only on the $1$ -sided CAD constants and $\vartheta $ ) such that

$$ \begin{align*} 2^{-k^*} \ell(Q) \leq \ell(I) \leq 2^{k^*}\ell(Q), &\quad \forall\,I \in \mathcal{W}_{Q}^{\vartheta, *}, \\ X(I) \rightarrow_{U_Q^{\vartheta}} X_Q, &\quad \forall\,I \in \mathcal{W}_{Q}^{\vartheta, *}, \\ \operatorname{dist}(I, Q) \leq 2^{k^*} \ell(Q), &\quad \forall\,I \in \mathcal{W}_{Q}^{\vartheta, *}. \end{align*} $$

Here, $X(I) \to _{U_Q^{\vartheta }} X_Q$ means that the interior of $U_Q^{\vartheta }$ 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 $\operatorname {dist}(Z, \partial \Omega ) \approx \operatorname {dist}(Z, \Omega \setminus U_Q^{\vartheta })$ with uniform control of implicit constants. The constant $k^*$ and the implicit constants in the condition $X(I) \to _{U_Q^{\vartheta }} X_Q$ depend at most on the allowable parameters on $\lambda $ and on $\vartheta $ . Moreover, given $I \in \mathcal {W}$ , we have that $I \in \mathcal {W}^{\vartheta ,*}_{Q_I}$ , where $Q_I \in \mathbb {D}$ 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 [Reference Hofmann and Martell31, Reference Hofmann, Martell and Toro35] for full details. We note, however, that in [Reference Hofmann and Martell31] the parameter $\vartheta $ is fixed. Here, we need to allow $\vartheta $ to depend on the aperture of the cones, and hence, it is convenient to include the superindex $\vartheta $ .

For a given $Q\in \mathbb {D}$ , the ‘Carleson box’ relative to Q is defined by

$$ \begin{align*}T_Q^{\vartheta}:=\operatorname{\mathrm{int}}\bigg(\bigcup_{Q'\in\mathbb{D}_Q}U_{Q'}^{\vartheta}\bigg). \end{align*} $$

For a given family $\mathcal {F}=\{Q_i\}\subset \mathbb {D}$ of pairwise disjoint cubes and a given $Q\in \mathbb {D}$ , we define the ‘local sawtooth region’ relative to $\mathcal {F}$ by

(2.11) $$ \begin{align} \Omega_{\mathcal{F},Q}^{\vartheta}:=\operatorname{\mathrm{int}}\bigg(\bigcup_{Q'\in\mathbb{D}_{\mathcal{F},Q}}U_{Q'}^{\vartheta}\bigg)=\operatorname{\mathrm{int}} \bigg(\bigcup_{I\in\mathcal{W}_{\mathcal{F},Q}^{\vartheta}} I^*\bigg), \end{align} $$

where $\mathcal {W}_{\mathcal {F},Q}^{\vartheta }:=\bigcup _{Q'\in \mathbb {D}_{\mathcal {F},Q}}\mathcal {W}_{Q'}^{\vartheta ,*}$ . Note that in the previous definition we may allow $\mathcal {F}$ to be empty in which case clearly $\Omega ^{\vartheta }_{{\tiny {\emptyset}} ,Q}=T_Q^{\vartheta }$ . Similarly, the ‘global sawtooth region’ relative to $\mathcal {F}$ is defined as

(2.12) $$ \begin{align} \Omega_{\mathcal{F}}^{\vartheta}:=\operatorname{\mathrm{int}}\bigg(\bigcup_{Q'\in\mathbb{D}_{\mathcal{F}}}U_{Q'}^{\vartheta}\bigg) = \operatorname{\mathrm{int}}\bigg(\bigcup_{I\in\mathcal{W}_{\mathcal{F}}^{\vartheta}}I^*\bigg), \end{align} $$

where $\mathcal {W}_{\mathcal {F}}^{\vartheta }:=\bigcup _{Q'\in \mathbb {D}_{\mathcal {F}}}\mathcal {W}_{Q'}^{\vartheta ,*}$ . If $\mathcal {F}$ is the empty set clearly $\Omega _{{\tiny {\emptyset}}}^{\vartheta }=\Omega $ . For a given $Q\in \mathbb {D}$ and $x\in \partial \Omega $ , let us introduce the ‘truncated dyadic cone’

$$\begin{align*}\Gamma_{Q}^{\vartheta}(x) := \bigcup_{x\in Q'\in\mathbb{D}_{Q}} U_{Q'}^{\vartheta}, \end{align*}$$

where it is understood that $\Gamma _{Q}^{\vartheta }(x)= \emptyset $ if $x\notin Q$ . Analogously, we can slightly fatten the Whitney boxes and use $I^{**}$ to define new fattened Whitney regions and sawtooth domains. More precisely, for every $Q\in \mathbb {D}$ ,

$$\begin{align*}T_Q^{\vartheta,*}:=\operatorname{\mathrm{int}}\bigg(\bigcup_{Q'\in\mathbb{D}_Q}U_{Q'}^{\vartheta,*}\bigg),\quad\Omega^{\vartheta,*}_{\mathcal{F},Q}:=\operatorname{\mathrm{int}}\bigg(\bigcup_{Q'\in\mathbb{D}_{\mathcal{F},Q}}U_{Q'}^{\vartheta,*}\bigg), \quad \Gamma^{\vartheta,*}_{Q}(x) := \bigcup_{x\in Q'\in\mathbb{D}_{Q_0}} U_{Q'}^{\vartheta,*}, \end{align*}$$

where $U_{Q}^{\vartheta ,*}:=\bigcup _{I\in \mathcal {W}_Q^{\vartheta ,*}}I^{**}$ . Similarly, we can define $T_Q^{\vartheta ,**}$ , $\Omega ^{\vartheta ,**}_{\mathcal {F},Q}$ , $\Gamma _Q^{\vartheta ,**}(x)$ , and $U^{\vartheta ,**}_{Q}$ by using $I^{***}$ in place of $I^{**}$ .

To define the ‘Carleson box’, $T_{\Delta }^{\vartheta }$ associated with a surface ball $\Delta =\Delta (x,r)$ , let $k(\Delta )$ denote the unique $k\in \mathbb {Z}$ such that $2^{-k-1}<200r\leq 2^{-k}$ and set

(2.13) $$ \begin{align} \mathbb{D}^{\Delta}:=\big\{Q\in\mathbb{D}_{k(\Delta)}:\:Q\cap 2\Delta\neq\emptyset\big\}. \end{align} $$

We then define

(2.14) $$ \begin{align} T_{\Delta}^{\vartheta}:=\operatorname{\mathrm{int}}\bigg(\bigcup_{Q\in\mathbb{D}^{\Delta}}\overline{T_Q^{\vartheta}}\bigg). \end{align} $$

We can also consider fattened versions of $T_{\Delta }^{\vartheta }$ given by

$$ \begin{align*} T_{\Delta}^{\vartheta,*}:=\operatorname{\mathrm{int}}\bigg(\bigcup_{Q\in\mathbb{D}^{\Delta}}\overline{T_Q^{\vartheta,*}}\bigg),\qquad T_{\Delta}^{\vartheta,**}:=\operatorname{\mathrm{int}}\bigg(\bigcup_{Q\in\mathbb{D}^{\Delta}}\overline{T_Q^{\vartheta,**}}\bigg). \end{align*} $$

Following [Reference Hofmann and Martell31, Reference Hofmann, Martell and Toro35], one can easily see that there exist constants $0<\kappa _1<1$ and $\kappa _0\geq 16\Xi $ (with $\Xi $ the constant in equation (2.6)), depending only on the allowable parameters and on $\vartheta $ , so that

(2.15) $$ \begin{align} &\kappa_1B_Q\cap\Omega\subset T_Q^{\vartheta}\subset T_Q^{\vartheta,*}\subset T_Q^{\vartheta,**}\subset \overline{T_Q^{\vartheta,**}}\subset\kappa_0B_Q\cap\overline{\Omega}=:\tfrac{1}{2}B_Q^*\cap\overline{\Omega}, \end{align} $$

(2.16) $$ \begin{align} & \ \tfrac{5}{4}B_{\Delta}\cap\Omega\subset T_{\Delta}^{\vartheta}\subset T_{\Delta}^{\vartheta,*}\subset T_{\Delta}^{\vartheta,**}\subset\overline{T_{\Delta}^{\vartheta,**}}\subset\kappa_0B_{\Delta}\cap\overline{\Omega}=:\tfrac{1}{2}B_{\Delta}^*\cap\overline{\Omega}, \end{align} $$

and also

(2.17) $$ \begin{align} Q\subset\kappa_0B_{\Delta}\cap\partial\Omega=\tfrac{1}{2}B_{\Delta}^*\cap\partial\Omega=:\tfrac{1}{2}\Delta^*,\qquad\forall\,Q\in\mathbb{D}^{\Delta}, \end{align} $$

where $B_Q$ is defined as in equation (2.7), $\Delta =\Delta (x,r)$ with $x\in \partial \Omega $ , $0<r<\operatorname {\mathrm {diam}}(\partial \Omega )$ and $B_{\Delta }=B(x,r)$ is so that $\Delta =B_{\Delta }\cap \partial \Omega $ . From our choice of the parameters, one also has that $B_Q^*\subset B_{Q'}^*$ whenever $Q\subset Q'$ .

Given Q we define the ‘localized dyadic conical square function’

(2.18) $$ \begin{align} \mathcal{S}_{Q}^{\vartheta} u(x):=\bigg(\iint_{\Gamma_{Q}^{\vartheta} (x)}|\nabla u(Y)|^2\delta(Y)^{1-n}\,dY\bigg)^{\frac12}, \qquad x\in \partial\Omega, \end{align} $$

for every $u\in W^{1,2}_{\mathrm {loc}} (T_{Q}^{\vartheta })$ . Note that $\mathcal {S}_{Q}^{\vartheta } u(x)=0$ for every $x\in \partial \Omega \setminus Q$ since $\Gamma _Q^{\vartheta } (x)= \emptyset $ in such case. The ‘localized dyadic nontangential maximal function’ is given by

(2.19) $$ \begin{align} \mathcal{N}_{Q}^{\vartheta} u(x) : = \sup_{Y\in \Gamma^{\vartheta,*}_{Q}(x)} |u(Y)|, \qquad x\in \partial\Omega, \end{align} $$

for every $u\in \mathscr {C}(T_{Q}^{\vartheta ,*})$ , where it is understood that $\mathcal {N}_{Q}^{\vartheta } u(x)= 0$ for every $x\in \partial \Omega \setminus Q$ .

Given $\alpha>0$ and $x\in \partial \Omega $ , we introduce the ‘cone with vertex at x and aperture $\alpha $ ’ defined as $\Gamma ^{\alpha }(x) = \{X \in \Omega : |X - x| \leq (1+\alpha ) \delta (X)\}$ . One can also introduce the ‘truncated cone’ for every $x\in \partial \Omega $ , and $0<r<\infty $ we set $\Gamma _{r}^{\alpha }(x) = B(x,r)\cap \Gamma ^{\alpha }(x)$ .

The ‘conical square function’ and the ‘nontangential maximal function’ are defined, respectively, as

(2.20) $$ \begin{align} \mathcal{S}^{\alpha} u(x):=\bigg(\iint_{\Gamma^{\alpha}(x)}|\nabla u(Y)|^2\delta(Y)^{1-n}\,dY\bigg)^{\frac12}, \qquad \mathcal{N}^{\alpha} u (x) := \sup_{X \in \Gamma^{\alpha} (x)} |u(X)|,\qquad x\in\partial\Omega \end{align} $$

for every $u\in W^{1,2}_{\mathrm {loc}}(\Omega )$ and $u\in \mathscr {C}(\Omega )$ , respectively. Analogously, the ‘truncated conical square function’ and the ‘truncated nontangential maximal function’ are defined, respectively, as

(2.21) $$ \begin{align} \mathcal{S}_{r}^{\alpha} u(x):=\bigg(\iint_{\Gamma_{r}^{\alpha}(x)}|\nabla u(Y)|^2\delta(Y)^{1-n}\,dY\bigg)^{\frac12}, \quad \mathcal{N}^{\alpha}_{r} u (x) := \sup_{X \in \Gamma^{\alpha}_{r} (x)} |u(X)|, \quad \end{align} $$

where $x\in \partial \Omega $ and $0<r<\infty $ , for every $u\in W^{1,2}_{\mathrm {loc}}(\Omega \cap B(x,r))$ and $u\in \mathscr {C}(\Omega \cap B(x,r))$ , respectively.

We would like to note that truncated dyadic cones are never empty. Indeed, in our construction, we have made sure that $X_Q\in U_Q^{\vartheta }$ for every $Q\in \mathbb {D}$ ; hence, for any $Q\in \mathbb {D}$ and $x\in Q$ one has $X_Q\in \Gamma _Q^{\vartheta }(x)$ . Moreover, $X_{Q'}\in \Gamma _Q^{\vartheta }(x)$ for every $Q'\in \mathbb {D}_Q$ with $Q'\ni x$ . For the regular truncated cones, it could happen that $\Gamma ^{\alpha }_r(x)= \emptyset $ unless $\alpha $ is sufficiently large. Suppose for instance that $\Omega =\{X=(x_1,\dots ,x_{n+1})\in \mathbb {R}^{n+1}: x_{1},\dots , x_{n+1}>0\}$ is the first orthant, then $\Gamma _{r}^{\alpha }(0)= \emptyset $ for any $0<r<\infty $ if $\alpha <\sqrt {n+1}-1$ . On the other hand, if $\alpha $ is sufficiently large, more precisely, if $\alpha \ge c_0^{-1}-1$ , where $c_0$ is the corkscrew constant (cf. Definition 2.1), then

(2.22) $$ \begin{align} X_{\Delta(x,r)}\in \Gamma_{r}^{\alpha}(x),\qquad \forall\,x\in\partial\Omega,\ 0<r<\operatorname{\mathrm{diam}}(\partial\Omega). \end{align} $$

4.1 Proof of $\mathrm {(a)}_{p'}\ \Longrightarrow \ \mathrm {(b)}_p$

Fix $\alpha>0$ and $N\ge 1$ . Take $\Delta _0 = \Delta (x_0, r_0)$ with $x_0 \in \partial \Omega $ and $0 < r_0 <\operatorname {\mathrm {diam}}(\partial \Omega )$ , and fix $f \in \mathscr {C}(\partial \Omega )$ with $\operatorname {\mathrm {supp}} f \subset N\Delta _0$ . We may assume that $N r_0<4\,\operatorname {\mathrm {diam}}(\partial \Omega )$ ; otherwise, $\partial \Omega $ is bounded and $4\operatorname {\mathrm {diam}}(\partial \Omega )/N\le r_0<\operatorname {\mathrm {diam}}(\partial \Omega )$ and we can work with $N'= 2\operatorname {\mathrm {diam}}(\partial \Omega )/r_0\in (2, N/2]$ and $N'\Delta _0=\partial \Omega $ .

Let u be the associated elliptic measure L-solution as in equation (3.4). Assume $\omega _L \in RH_{p'}(\partial \Omega , \omega _{L_0})$ , and our goal is to obtain that equation (3.5) holds. By Gehring’s lemma [Reference Gehring26] (see also [Reference Coifman and Fefferman11]), there exists $s>1$ such that $\omega _{L}\in RH_{p's} (\partial \Omega ,\omega _{L_0})$ .

Introduce the family of pairwise disjoint cubes

$$\begin{align*}\mathcal{F}_{\Delta_0}:=\{Q\in\mathbb{D}(\partial\Omega): (N+3\Xi)r_0<\ell(Q)\le 2(N+3\Xi) r_0,\ Q\cap 3\Xi\Delta_0\neq\emptyset\}. \end{align*}$$

Take $x \in \Delta _0$ and $X \in \Gamma _{r_0}^{\alpha }(x)$ . Let $I_X \in \mathcal {W}$ be such that $X \in I_X$ . Take $y_X \in \partial \Omega $ such that $\operatorname {dist}(I_X, \partial \Omega ) = \operatorname {dist}(I_X, y_X)$ , and let $Q_X \in \mathbb {D}$ be the unique dyadic cube satisfying $\ell (Q_X) = \ell (I_X)$ and $y_X \in Q_X$ . By construction (see Section 2.3), $I_X \in \mathcal {W}_{Q_X}^{\vartheta ,*}$ and thus $I^* \subset \Gamma _{Q_X}(y_X)$ . Thus, by the properties of the Whitney cubes

$$\begin{align*}\delta(X) \le |X-y_X| \le \operatorname{\mathrm{diam}}(I_X)+\operatorname{dist}(I_X, y_X) \le \frac54\operatorname{dist}(I_X, \partial\Omega) \le \frac54\delta(X) \end{align*}$$

and

$$ \begin{align*} 4\ell(Q_X) = 4\ell(I_X) \le \operatorname{dist}(I_X,\partial\Omega) \le \delta(X) \le \frac54\operatorname{dist}(I_X, \partial\Omega) \le 50\sqrt{n+1}\ell(I_X) = 50\sqrt{n+1}\ell(Q_X). \end{align*} $$

These and the fact that $X \in \Gamma _{r_0}^{\alpha }(x)$ give

$$\begin{align*}\ell(Q_X) < \frac1{4}\,\delta(X) \le \frac14 |X-x| < \frac14r_0. \end{align*}$$

Also, for every $z\in Q_X$

$$\begin{align*}|z-x_0| \le |z-y_X|+|y_X-X|+|X-x|+|x-x_0| < 2\Xi\ell(Q_X)+ \frac94 |X-x| +r_0 < (\Xi+ 4)\,r_0 \le 3\Xi r_0, \end{align*}$$

since $\Xi \ge 2$ , and

$$\begin{align*}|z-x| \le |z-y_X|+|y_X-X|+ |X-x| < 2\Xi\ell(Q_X)+(3+\alpha)\delta(X) < (2\Xi+\alpha) \delta(X) =:C_{\alpha}\delta(X) \end{align*}$$

since $X \in \Gamma _{r_0}^{\alpha }(x)$ . Thus, $Q_X\subset 3\Xi \Delta _0\cap \Delta (x, C_{\alpha }\delta (X))$ and there exists a unique $\widetilde {Q}_X\in \mathcal {F}_{\Delta _0}$ such that $Q_X\subsetneq \widetilde {Q}_X$ . In particular, $X\in I_X\subset U_{Q_X}\subset \Gamma _{\widetilde {Q}_X}(y)$ for all $y \in Q_X$ and

$$ \begin{align*} |u(X)| \leq \mathcal{N}_{\widetilde{Q}_X} u (y), \qquad \text{for all } y \in Q_X. \end{align*} $$

Taking the average over $Q_X$ with respect to $\omega _{L_0}^{X_{\Delta _0}}$ , we arrive at

where in the last inequality we have used that $\delta (X)\le |X-x|< r_0$ since $\Gamma _{r_0}^{\alpha }(x)\subset B(x,r_0)$ . Taking now the supremum over all $X \in \Gamma _{r_0}^{\alpha }(x)$ , we arrive at

Applying the Hardy–Littlewood maximal inequality and the fact that the set $\mathcal {F}_{\Delta _0}$ has bounded cardinality, we have

(4.8)

where we have used that for every $Q \in \mathcal {F}_{\Delta _0}$ we have $\operatorname {\mathrm {supp}} (\mathcal {N}_Q u)\subset Q$ .

Let us also observe that for every $Q\in \mathcal {F}_{\Delta _0}$ we can pick $y_Q\in Q\cap 3\Xi \Delta _0$ so that if $z\in N\Delta _0$ there holds

$$\begin{align*}|z-x_Q| \le |z-x_0|+|x_0-y_Q|+|y_Q-x_Q| \le (N+3\Xi) r_0+\Xi r_Q <2\Xi r_Q. \end{align*}$$

That is, $N\Delta _0\subset 2\widetilde {\Delta }_{Q}$ , and we are now ready to invoke [Reference Akman, Hofmann, Martell and Toro1, Proposition 2.57] to see that

(4.9)

To continue let $x\in Q\in \mathcal {F}_{\Delta _0}$ , and let $\Delta $ be a surface ball such that $x \in \Delta $ and $0 < r_{\Delta } < 4\Xi r_Q$ . In particular, $\Delta \subset C_N\Delta _0=\widetilde {\Delta }_0$ and $Q\subset \widetilde {\Delta }_0$ . Note that $\omega _{L_0}^{X_{\Delta _0}}\approx _N \omega _{L_0}^{X_{\widetilde {\Delta }_0}}$ by Harnack’s inequality and the fact that $\delta (X_{\Delta _0})\approx r_0$ , $\delta (X_{\widetilde {\Delta }_0})\approx _N r_0$ and $|X_{\Delta _0}-X_{\widetilde {\Delta }_0}|\lesssim _N r_0$ .

Recall that $\omega _L \in RH_{p' s}(\partial \Omega , \omega _{L_0})$ implies $\omega _L \in RH_{p' s}(\widetilde {\Delta }_0, \omega ^{X_{\widetilde {\Delta }_0}}_{L_0})$ (uniformly). Therefore, using Hölder’s inequality and recalling that $h(\cdot; L, L_0, X)$ denotes the Radon–Nikodym derivative of $\omega _L^X$ with respect to $\omega _{L_0}^X$ , we get

This, equation (4.9) and equation (4.8) yield

where we have used the boundedness of the local Hardy–Littlewood maximal function in the second term on $L^{\frac {p}{(p's)'}}(\widetilde {\Delta }_0,\omega _{L_0}^{X_{\widetilde {\Delta }_0}})$ , which follows from $p>(p's)'$ and the fact that $\omega _{L_0}^{X_{\widetilde {\Delta }_0}}$ is doubling in $10\widetilde {\Delta }_0$ . This completes the proof of $\mathrm {(b)}_p$ . $\Box $

4.2 Proof of $\mathrm {(b)}_p\ \Longrightarrow \ \mathrm {(a)}_{p'}$

Fix $p\in (1,\infty )$ , and assume that L is $L^{p}(\omega _{L_0})$ -solvable. That is, for some fixed $\alpha _0$ and some $N\ge 1$ , there exists $C_{\alpha _0, N}\ge 1$ (depending only on n, the $1$ -sided NTA constants, the CDC constant, the ellipticity of $L_0$ and L, $\alpha _0$ , N and p) such that equation (3.5) holds for u as in equation (3.4) for any $f\in \mathscr {C}(\partial \Omega )$ with $\operatorname {\mathrm {supp}} f\subset N\Delta _0$ . From this and equation (4.6), we conclude that we can assume that $\alpha \ge c_0^{-1}-1$ , where $c_0$ is the corkscrew constant (cf. Definition 2.1), and we have

(4.10) $$ \begin{align} \|\mathcal{N}^{\alpha}_{r_0} u\|_{L^p(\Delta_0, \omega_{L_0}^{X_{\Delta_0}})} \lesssim_{\alpha,\alpha_0} \|\mathcal{N}^{\alpha_0}_{4r_0} u\|_{L^p(4\Delta_0, \omega_{L_0}^{X_{4\Delta_0}})} \leq C_{\alpha_0,N} \|f\|_{L^p(N\Delta_0, \omega_{L_0}^{X_{\Delta_0}})}, \end{align} $$

for u as in equation (3.4) with $f\in \mathscr {C}(\partial \Omega )$ with $\operatorname {\mathrm {supp}} f\subset N\Delta _0$ and for any $\Delta _0 = \Delta (x_0,r_0)$ , $x_0 \in \partial \Omega $ and $0 < r_0 < \operatorname {\mathrm {diam}}(\partial \Omega )/4$ . It is routine to see this estimate also holds with $r_0\approx \operatorname {\mathrm {diam}}(\partial \Omega )$ . Indeed, by splitting f into its positive and negative parts we may assume that $f\ge 0$ . In that case, if $x\in \partial \Omega $ and $X\in \Gamma ^{\alpha }_{r_0}(x)\setminus \Gamma ^{\alpha }_{\operatorname {\mathrm {diam}}(\partial \Omega )/5}(x)$ , we have that $\delta (X)\approx \operatorname {\mathrm {diam}}(\partial \Omega )$ , and by equation (2.22), one has that $X':=X_{\Delta (x,\operatorname {\mathrm {diam}}(\partial \Omega )/5)}\in \Gamma ^{\alpha }_{\operatorname {\mathrm {diam}}(\partial \Omega )/5}(x)$ . Harnack’s inequality implies then that $u(X)\approx u(X')$ , and this shows that $\mathcal {N}^{\alpha }_{r_0} u(x)\lesssim \mathcal {N}^{\alpha }_{\operatorname {\mathrm {diam}}(\partial \Omega )/5} u(x)$ . Further details are left to the interested reader.

We claim that, for every $\Delta _0 = \Delta (x_0,r_0)$ , $x_0 \in \partial \Omega $ and $0 < r_0 < \operatorname {\mathrm {diam}}(\partial \Omega )$ , and for every $ f\in \mathscr {C}(\partial \Omega )$ with $\operatorname {\mathrm {supp}} f\subset N\Delta _0$