Harmonic measure and quantitative connectivity: geometric characterization of the $L^p$-solvability of the Dirichlet problem

It is well-known that quantitative, scale invariant absolute continuity (more precisely, the weak-$A_\infty$ property) of harmonic measure with respect to surface measure, on the boundary of an open set $ \Omega\subset \mathbb{R}^{n+1}$ with Ahlfors-David regular boundary, is equivalent to the solvability of the Dirichlet problem in $\Omega$, with data in $L^p(\partial\Omega)$ for some $p<\infty$. In this paper, we give a geometric characterization of the weak-$A_\infty$ property, of harmonic measure, and hence of solvability of the $L^p$ Dirichlet problem for some finite $p$. This characterization is obtained under background hypotheses (an interior corkscrew condition, along with Ahlfors-David regularity of the boundary) that are in the nature of best possible: we provide counter-examples in the absence of either of them (or even one of the two, upper or lower, Ahlfors-David bounds).


Introduction
A classical criterion of Wiener characterizes the domains in which one can solve the Dirichlet problem for Laplace's equation with continuous boundary data, and with continuity of the solution up to the boundary. In this paper, we address the analogous issue in the case of singular data. To be more precise, the present work provides a purely geometric characterization of the open sets for which L p solvability holds, for some p < ∞, and with non-tangential convergence to the data a.e., thus allowing for singular boundary data. We establish this characterization in the presence of background hypotheses (an interior corkscrew condition [see Definition 2.5 below], and Ahlfors-David regularity of the boundary [Definition 2.1]) that are in the nature of best possible, in the sense that there are counter-examples in the absence of either of them (or of even one of the two, upper or lower, Ahlfors- David bounds).
Solvability of the L p Dirichlet problem is fundamentally tied to quantitative absolute continuity of harmonic measure with respect to surface measure on the boundary: indeed, it is equivalent to the so-called "weak-A ∞ " property of the harmonic measure (see Definition 2.16). It is through this connection to quantitative absolute continuity of harmonic measure that we shall obtain our geometric characterization of L p solvability.
The study of the relationship between the geometry of a domain, and absolute continuity properties of its harmonic measure, has a long history. A classical result of F. and M. Riesz [RR] states that for a simply connected domain Ω in the complex plane, rectifiability of ∂Ω implies that harmonic measure for Ω is absolutely continuous with respect to arclength measure on the boundary. A quantitative version of this theorem was later proved by Lavrentiev [Lav]. More generally, if only a portion of the boundary is rectifiable, Bishop and Jones [BJ] have shown that harmonic measure is absolutely continuous with respect to arclength on that portion. They also present a counter-example to show that the result of [RR] may fail in the absence of some connectivity hypothesis (e.g., simple connectedness).
In dimensions greater than 2, a fundamental result of Dahlberg [Dah] establishes a quantitative version of absolute continuity, namely that harmonic measure belongs to the class A ∞ in an appropriate local sense (see Definitions 2.16 and 2.20 below), with respect to surface measure on the boundary of a Lipschitz domain.
The result of Dahlberg was extended to the class of Chord-arc domains (see Definition 2.8) by David and Jerison [DJ], and independently by Semmes [Sem]. The Chord-arc hypothesis was weakened to that of a two-sided corkscrew condition (Definition 2.5) by Bennewitz and Lewis [BL], who then drew the conclusion that harmonic measure is weak-A ∞ (in an appropriate local sense, see Definitions 2. 16 and 2.20) with respect to surface measure on the boundary; the latter condition is similar to the A ∞ condition, but without the doubling property, and is the best conclusion that can be obtained under the weakened geometric conditions considered in [BL]. We note that weak-A ∞ is still a quantitative, scale invariant version of absolute continuity.
More recently, one of us (Azzam) has given in [Azz] a geometric characterization of the A ∞ property of harmonic measure with respect to surface measure for domains with n-dimensional Ahlfors-David regular (n-ADR) boundary (see Definition 2.1). Azzam's results are related to those of the present paper, so let us describe them in a bit more detail. Specifically, he shows that for a domain Ω with n-ADR boundary, harmonic measure is in A ∞ with respect to surface measure, if and only if 1) ∂Ω is uniformly rectifiable (n-UR) 1 , and 2) Ω is semi-uniform in the sense of Aikawa and Hirata [AH]. The semi-uniform condition is a connectivity condition which states that for some uniform constant M, every pair of points x ∈ Ω and y ∈ ∂Ω may be connected by a rectifiable curve γ = γ(y, x), with γ \ {y} ⊂ Ω, with length ℓ(γ) ≤ M|x − y|, and which satisfies the "cigar path" condition (1.1) min ℓ γ(y, z) , ℓ γ(z, x) ≤ M dist(z, ∂Ω) , ∀ z ∈ γ .
Semi-uniformity is a weak version of the well known uniform condition, whose definition is similar, except that it applies to all pairs of points x, y ∈ Ω. For example, the unit disk centered at the origin, with the slit {−1/2 ≤ x ≤ 1/2, y = 0} removed, is semi-uniform, but not uniform. It was shown in [AH] that for a domain satisfying a John condition and the Capacity Density Condition (in particular, for a domain with an n-ADR boundary), semi-uniformity characterizes the doubling property of harmonic measure. The method of [Azz] is, broadly speaking, related to that of [DJ], and of [BL]. In [DJ], the authors show that a Chord-arc domain Ω may be approximated in a "Big Pieces" sense (see [DJ] or [BL] for a precise statement; also cf. Definition 2.13 below) by Lipschitz subdomains Ω ′ ⊂ Ω; this fact allows one to reduce matters to the result of Dahlberg via the maximum principle (a method which, to the present authors' knowledge, first appears in [JK] in the context of BMO 1 domains). The same strategy, i.e., Big Piece approximation by Lipschitz subdomains, is employed in [BL]. Similarly, in [Azz], matters are reduced to the result of [DJ], by showing that for a domain Ω with an n-ADR boundary, Ω is semi-uniform with a uniformly rectifiable boundary if and only if it has "Very Big Pieces" of Chord-arc subdomains (see [Azz] for a precise statement of the latter condition). As mentioned above, the converse direction is also treated in [Azz]. In that case, given an interior corkscrew condition (which holds automatically in the presence of the doubling property of harmonic measure), and provided that ∂Ω is n-ADR, the A ∞ (or even weak-A ∞ ) property of harmonic measure was already known to imply uniform rectifiability of the boundary [HM3] (although the published version appears in [HLMN]; see also [MT] for an alternative proof, and a somewhat more general result); as in [AH], semi-uniformity follows from the doubling property, although in [Azz], the author manages to show this while dispensing with the John domain background assumption (given a harmlessly strengthened version of the doubling property).
Thus, in [Azz], the connectivity condition (semi-uniformity), is tied to the doubling property of harmonic measure, and not to absolute continuity. On the other hand, in light of the example of [BJ], and on account of the aforementioned connection to solvability of the Dirichlet problem, it has been an important open problem to determine the minimal connectivity assumption which, in conjunction with uniform rectifiability of the boundary, yields quantitative absolute continuity of harmonic measure with respect to surface measure. In the present work, we present a connectivity condition, significantly milder than semi-uniformity, which we call the weak local John condition (see Definition 2.13 below), and which solves this problem. Thus, we obtain a geometric characterization of the domains for which one has quantitative absolute continuity of harmonic measure; equivalently, for which one has solvability of the Dirichlet problem with singular (L p ) data (see Theorem 1.3 below). In fact, we provide two geometric characterizations of such domains, one in terms of uniform rectifiability combined with the weak local John condition, the other in terms of approximation of the boundary in a big pieces sense, by boundaries of Chord-arc subdomains.
Let us now describe the weak local John condition, which says, roughly speaking, that from each point x ∈ Ω, there is local non-tangential access to an ample portion of a surface ball at a scale on the order of δ Ω (x) := dist(x, ∂Ω). Let us make this a bit more precise. A "carrot path" (aka non-tangential path) joining a point x ∈ Ω, and a point y ∈ ∂Ω, is a connected rectifiable path γ = γ(y, x), with endpoints y and x, such that for some λ ∈ (0, 1) and for all z ∈ γ, where ℓ γ(y, z) denotes the arc-length of the portion of the original path with endpoints y and z. For x ∈ Ω, and N ≥ 2, set ∆ x = ∆ N x := B x, Nδ Ω (x) ∩ ∂Ω . We assume that every point x ∈ Ω may be joined by a carrot path to each y in a "Big Piece" of ∆ x , i.e., to each y in a Borel subset F ⊂ ∆ x , with σ(F) ≥ θσ(∆ x ), where σ denotes surface measure on ∂Ω, and where the parameters N ≥ 2, λ ∈ (0, 1), and θ ∈ (0, 1] are uniformly controlled. We refer to this condition as a "weak local John condition", although "weak local semi-uniformity" would be equally appropriate. See Definitions 2.9, 2.11 and 2.13 for more details. We remark that a strong version of the local John condition (i.e., with θ = 1) has appeared in [HMT], in connection with boundary Poincaré inequalities for non-smooth domains.
The main result in the present work is the following geometric characterization of quantitative absolute continuity of harmonic measure, and of the L p solvability of the Dirichlet problem. The terminology used here will be defined in the sequel. Theorem 1.3. Let Ω ⊂ R n+1 , n ≥ 1, be an open set satisfying an interior corkscrew condition (see Definition 2.5 below), and suppose that ∂Ω is n-dimensional Ahlfors-David regular (n-ADR; see Definition 2.1 below). Then the following are equivalent: (1) ∂Ω is Uniformly Rectifiable (n-UR; see Definition 2.3 below) and Ω satisfies the weak local John condition (see Definition 2.13 below). (

2) Ω satisfies an Interior Big Pieces of Chord-Arc Domains (IBPCAD) condition (see Definition 2.14 below). (3) Harmonic measure ω is locally in weak-A ∞ (see Definition 2.20 below) with
respect to surface measure σ on ∂Ω. (4) The L p Dirichlet problem is solvable for some p < ∞, i.e., for some p < ∞, there is a constant C such that if g ∈ L p (∂Ω), then the solution to the Dirichlet problem with data g, is well defined as u(x) := ∂Ω gdω x for each x ∈ Ω, converges to g non-tangentially, and enjoys the estimate where N * u is a suitable version of the non-tangential maximal function of u.
Some explanatory comments are in order. The proof has two main new ingredients: the implication (1) implies (2), and the fact that the weak-A ∞ property of harmonic measure implies the weak local John condition (this is the new part of (3) implies (1)). In turn, we split these main new results into two theorems: the first implication is the content of Theorem 1.5 below, and the second is the content of Theorem 1.6. We remark that the interior corkscrew condition is not needed for (1) implies (2) (nor for (2) implies (3) if and only if (4)). Rather, it is crucial for (3) implies (1) (see Appendix A).
As regards the other implications, the fact that (2) implies (3) follows by a wellknown argument using the maximum principle and the result of [DJ] and [Sem] for Chord-arc domains 2 , along with the criterion for weak-A ∞ obtained in [BL]; the equivalence of (3) and (4) is well known, and we refer the reader to, e.g., [HLe,Section 4], and to [H] for details. The implication (3) implies (1) has two parts. As mentioned above, the fact that weak-A ∞ implies weak local John is new, and is the content of Theorem 1.6. The remaining implication, namely that weak-A ∞ implies n-UR, is the main result of [HM3]; an alternative proof, with a more general result, appears in [MT], and see also [HLMN] for the final published version of the results of [HM3], along with an extension to the p-harmonic setting.
We note that our background hypotheses (upper and lower n-ADR, and interior corkscrew) are in the nature of best possible: one may construct a counter-example in the absence of any one of them, for at least one direction of this chain of implications, as we shall discuss in Appendix A.
As explained above, the main new contributions of the present work are contained in the following pair of theorems, Theorem 1.5. Let Ω ⊂ R n+1 , n ≥ 1, be an open set, not necessarily connected, with an n-dimensional Ahlfors-David regular (n-ADR) boundary. Then the following are equivalent: (i) ∂Ω is uniformly rectifiable (n-UR), and Ω satisfies the weak local John condition.

(ii) Ω satisfies an Interior Big Pieces of Chord-Arc Domains (IBPCAD) condition.
Only the direction (i) implies (ii) is new. For the converse, the fact that IBPCAD implies the weak local John condition is immediate from the definitions. Moreover, the boundary of a Chord-arc domain is n-UR, and an n-ADR set with big pieces of n-UR is also n-UR (see [DS2]). As noted above, that (ii) implies the weak-A ∞ property follows by well known arguments. Theorem 1.6. Let Ω ⊂ R n+1 , n ≥ 1, be an open set satisfying an interior corkscrew condition and suppose that ∂Ω is n-dimensional Ahlfors- David regular (n-ADR). If the harmonic measure for Ω satisfies the weak-A ∞ condition, then Ω satisfies the weak local John condition.
Let us mention that the present paper is a combination of unpublished work of two different subsets of the present authors: Theorem 1.5 is due to the second and third authors, and was first posted in the draft manuscript [HM5] 3 ; Theorem 1.6 is due to the first, fourth and fifth authors, and appeared first in the draft manuscript [AMT2].
The paper is organized as follows. In the next section, we set notation and give some definitions. In Part 1 of the paper (Sections 3-8), we give the proof of Theorem 1.5. In Part 2 of the paper (Sections 9-16) we give the proof of Theorem 1.6. Finally, in Appendix A, we discuss some counter-examples which show that our background hypotheses are in the nature of best possible.

Notation and definitions
• Unless otherwise stated, 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 b, a b, and a ≈ b to mean, respectively, that a ≤ Cb, a ≥ cb, and 0 < c ≤ a/b ≤ C, where the constants c and C are as above, unless explicitly noted to the contrary. In some occasions we will employ the notation a λ b, a λ b and a ≈ λ b to emphasize that the previous implicit constants c and/or C may depend on some relevant parameter λ. 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.
• Ω will always denote an open set in R n+1 , not necessarily connected unless otherwise specified.
• We use the notation γ(x, y) to denote a rectifiable path with endpoints x and y, and its arc-length will be denoted ℓ(γ(x, y)). Given such a path, if z ∈ γ(x, y), we use the notation γ(z, y) to denote the portion of the original path with endpoints z and y.
• The open (n + 1)-dimensional Euclidean ball of radius r will be denoted B (x, r).
• Given a Euclidean ball B or surface ball ∆, its radius will be denoted r B or r ∆ , respectively.
3 An earlier version of this work [HM4] gave a direct proof of the fact that (1) implies (3) in Theorem 1.3, without passing through condition (2).
• We let H n denote n-dimensional Hausdorff measure, and let σ := H n ∂Ω denote the surface measure on ∂Ω.
• For a Borel set A ⊂ R n+1 , we let χ A denote the usual indicator function of A, i.e.
• For a Borel set A ⊂ R n+1 , we let int(A) denote the interior of A.
• Given a Borel measure µ, and a Borel set A, with positive and finite µ measure, we set A f dµ := µ(A) −1 A f dµ. • We shall use the letter I (and sometimes J) to denote a closed (n + 1)-dimensional Euclidean dyadic cube with sides parallel to the co-ordinate axes, and we let ℓ(I) denote the side length of I. If ℓ(I) = 2 −k , then we set k I := k. Given an n-ADR set E ⊂ R n+1 , we use Q (or sometimes P or R) to denote a dyadic "cube" on E.
Definition 2.1. (n-ADR) (aka n-Ahlfors- David regular). We say that a set E ⊂ R n+1 , of Hausdorff dimension n, is n-ADR if it is closed, and if there is some uniform constant C such that where diam(E) may be infinite. Here, ∆(x, r) := E ∩ B(x, r) is the surface ball of radius r, and as above, σ := H n ⌊ E is the "surface measure" on E.
Definition 2.3. (n-UR) (aka n-uniformly rectifiable). An n-ADR (hence closed) set E ⊂ R n+1 is n-UR if and only if it contains "Big Pieces of Lipschitz Images" of R n ("BPLI"). This means that there are positive constants c 1 and C 1 , such that for each x ∈ E and each r ∈ (0, diam(E)), there is a Lipschitz mapping ρ = ρ x,r : R n → R n+1 , with Lipschitz constant no larger than C 1 , such that We recall that n-dimensional rectifiable sets are characterized by the property that they can be covered, up to a set of H n measure 0, by a countable union of Lipschitz images of R n ; we observe that BPLI is a quantitative version of this fact.
We remark that, at least among the class of n-ADR sets, the n-UR sets are precisely those for which all "sufficiently nice" singular integrals are L 2 -bounded [DS1]. In fact, for n-ADR sets in R n+1 , the L 2 boundedness of certain special singular integral operators (the "Riesz Transforms"), suffices to characterize uniform rectifiability (see [MMV] for the case n = 1, and [NTV] in general). We further remark that there exist sets that are n-ADR (and that even form the boundary of a domain satisfying interior corkscrew and Harnack Chain conditions), but that are totally non-rectifiable (e.g., see the construction of Garnett's "4-corners Cantor set" in [DS2,Chapter 1]). Finally, we mention that there are numerous other characterizations of n-UR sets (many of which remain valid in higher co-dimensions); cf. [DS1,DS2].
Definition 2.4. ("UR character"). Given an n-UR set E ⊂ R n+1 , its "UR character" is just the pair of constants (c 1 , C 1 ) involved in the definition of uniform rectifiability, along with the ADR constant; or equivalently, the quantitative bounds involved in any particular characterization of uniform rectifiability.
Definition 2.5. (Corkscrew condition). Following [JK], we say that an open set Ω ⊂ R n+1 satisfies the corkscrew condition if for some uniform constant c > 0 and for every surface ball ∆ := ∆(x, r), with x ∈ ∂Ω and 0 < r < diam(∂Ω), there is a ball B(x ∆ , cr) ⊂ B(x, r) ∩ Ω. The point x ∆ ⊂ Ω is called a corkscrew point relative to ∆. We note that we may allow r < C diam(∂Ω) for any fixed C, simply by adjusting the constant c. In order to emphasize that B(x ∆ , cr) ⊂ Ω, we shall sometimes refer to this property as the interior corkscrew condition.
Definition 2.6. (Harnack Chains, and the Harnack Chain condition [JK]). Given two points x, x ′ ∈ Ω, and a pair of numbers M, N ≥ 1, an (M, N)-Harnack Chain connecting x to x ′ , is a chain of open balls B 1 , . . . , . We say that Ω satisfies the Harnack Chain condition if there is a uniform constant M such that for any two points x, x ′ ∈ Ω, there is an (M, N)-Harnack Chain connecting them, with N depending only on M and the ratio |x − x ′ |/ min δ Ω (x), δ Ω (x ′ ) .
Definition 2.7. (NTA). Again following [JK], we say that a domain Ω ⊂ R n+1 is NTA (Non-tangentially accessible) if it satisfies the Harnack Chain condition, and if both Ω and Ω ext := R n+1 \ Ω satisfy the corkscrew condition.
Definition 2.8. (CAD). We say that a connected open set Ω ⊂ R n+1 is a CAD (Chord-arc domain), if it is NTA, and if ∂Ω is n-ADR.
Definition 2.9. (Carrot path). Let Ω ⊂ R n+1 be an open set. Given a point x ∈ Ω, and a point y ∈ ∂Ω, we say that a connected rectifiable path γ = γ(y, x), with endpoints y and x, is a carrot path (more precisely, a λ-carrot path) connecting y to x, if γ \ {y} ⊂ Ω, and if for some λ ∈ (0, 1) and for all z ∈ γ, With a slight abuse of terminology, we shall sometimes refer to such a path as a λ-carrot path in Ω, although of course the endpoint y lies on ∂Ω.
A carrot path is sometimes referred to as a non-tangential path.
Thus, a weak local John point is non-tangentially connected to an ample portion of the boundary, locally. We observe that one can always choose N smaller, for possibly different values of θ and λ, by moving from x to a point x ′ on a line segment joining x to the boundary.
Remark 2.12. We observe that it is a slight abuse of notation to write ∆ x , since the latter is not centered on ∂Ω, and thus it is not a true surface ball; on the other hand, there are true surface balls, ∆ ′ x := ∆(x, (N − 1)δ Ω (x)) and ∆ ′′ x := ∆(x, (N + 1)δ Ω (x)), centered at a "touching point"x ∈ ∂Ω with δ Ω (x) = |x −x|, which, respectively, are contained in, and contain, ∆ x .
Definition 2.14. (IBPCAD). We say that a connected open set Ω ⊂ R n+1 has Interior Big Pieces of Chord-Arc Domains (IBPCAD) if there exist positive constants η and C, and N ≥ 2, such that for every x ∈ Ω, with δ Ω (x) < diam(∂Ω), there is a Chord-arc domain Ω x ⊂ Ω satisfying Remark 2.15. In the presence of an interior corkscrew condition, Definition 2.14 is easily seen to be essentially equivalent to the following more standard "Big Pieces" condition: there are positive constants η and C (perhaps slightly different to that in Definition 2.14), such that for each surface ball ∆ := ∆(x, r) = B(x, r) ∩ ∂Ω, x ∈ ∂Ω and r < diam(∂Ω), and for any corkscrew point x ∆ relative to ∆ there is a Chord-arc domain Ω ∆ satisfying • The Chord-arc constants of the domains Ω ∆ are uniform in ∆.
Definition 2.16. (A ∞ , weak-A ∞ , and weak-RH q ). Given an n-ADR set E ⊂ R n+1 , and a surface ball ∆ 0 := B 0 ∩ E centered on E, we say that a Borel measure µ defined on E belongs to A ∞ (∆ 0 ) if there are positive constants C and s such that for each Similarly, we say that µ ∈ weak-A ∞ (∆ 0 ) if for each surface ball ∆ = B ∩ E centered on E, with 2B ⊆ B 0 , We recall that, as is well known, the condition µ ∈ weak-A ∞ (∆ 0 ) is equivalent to the property that µ ≪ σ in ∆ 0 , and that for some q > 1, the Radon-Nikodym derivative k := dµ/dσ satisfies the weak reverse Hölder estimate with B centered on E. We shall refer to the inequality in (2.19) as a "weak-RH q " estimate, and we shall say that k ∈ weak-RH q (∆ 0 ) if k satisfies (2.19).
Suppose that E ⊂ R n+1 is an n-ADR set. Then there exist constants a 0 > 0, s > 0 and C 1 < ∞, depending only on n and the ADR constant, such that for each k ∈ Z, there is a collection of Borel sets ("cubes") for all k, j and for all ϑ ∈ (0, a 0 ).
A few remarks are in order concerning this lemma.
• In the setting of a general space of homogeneous type, this lemma has been proved by Christ [Chr] (see also [HK]), with the dyadic parameter 1/2 replaced by some constant δ ∈ (0, 1). In fact, one may always take δ = 1/2 (see [HMMM, Proof of Proposition 2.12]). In the presence of the Ahlfors-David property (2.2), the result already appears in [DS1,DS2]. Some predecessors of this construction have appeared in [Da1] and [Da2].
• For our purposes, we may ignore those k ∈ Z such that 2 −k diam(E), in the case that the latter is finite.
• We shall denote by D = D(E) the collection of all relevant Q k j , i.e., where, if diam(E) is finite, the union runs over those k such that 2 −k diam(E).
• Properties (iv) and (v) imply that for each cube Q ∈ D k , there is a point x Q ∈ E, a Euclidean ball B(x Q , r Q ) and a surface ball for some uniform constant C. We shall refer to the point x Q as the "center" of Q.
• For a dyadic cube Q ∈ D k , we shall set ℓ(Q) = 2 −k , and we shall refer to this quantity as the "length" of Q. Evidently, by adjusting if necessary some parameters, we can assume that diam(Q) ≤ ℓ(Q) diam(Q). We shall denote (2.25) • For a dyadic cube Q ∈ D, we let k(Q) denote the dyadic generation to which Q belongs, i.e., we set k = k(Q) if Q ∈ D k ; thus, ℓ(Q) = 2 −k(Q) .
• Given R ∈ D, we set For j ≥ 1, we also let • For a pair of cubes Q ′ , Q ∈ D, if Q ′ is a dyadic child of Q, i.e., if Q ′ ⊂ Q, and ℓ(Q) = 2ℓ(Q ′ ), then we write Q ′ ⊳ Q.
• For λ > 1, we write With the dyadic cubes in hand, we may now define the notion of a corkscrew point relative to a cube Q.
Definition 2.28. (Corkscrew point relative to Q). Let Ω satisfy the corkscrew condition (Definition 2.5), suppose that ∂Ω is n-ADR, and let Q ∈ D(∂Ω). A corkscrew point relative to Q is simply a corkscrew point relative to the surface ball ∆(x Q , r Q ) defined in (2.24).
Definition 2.29. (Coherency and Semi-coherency). [DS2]. Let E ⊂ R n+1 be an n-ADR set. Let T ⊂ D(E). We say that T is coherent if the following conditions hold: (a) T contains a unique maximal element Q(T) which contains all other elements of T as subsets. (b) If Q belongs to T, and if Q ⊂ Q ⊂ Q(T), then Q ∈ T. (c) Given a cube Q ∈ T, either all of its children belong to T, or none of them do.
We say that T is semi-coherent if conditions (a) and (b) hold. We shall refer to a coherent or semi-coherent collection T as a tree.
Part 1: Proof of Theorem 1.5 3. Preliminaries for the Proof of Theorem 1.5 We begin by recalling a bilateral version of the David-Semmes "Corona decomposition" of an n-UR set. We refer the reader to [HMM] for the proof.
Lemma 3.1. ( [HMM,Lemma 2.2]) Let E ⊂ R n+1 be an n-UR set. Then given any positive constants η ≪ 1 and K ≫ 1, there is a disjoint decomposition D(E) = G∪B, satisfying the following properties.
(1) The "Good" collection G is further subdivided into disjoint trees, such that each such tree T is coherent (Definition 2.29).
(2) The "Bad" cubes, as well as the maximal cubes Q(T), T ⊂ G, satisfy a Carleson packing condition: (3) For each T ⊂ G, there is a Lipschitz graph Γ T , with Lipschitz constant at most η, such that, for every Q ∈ T, where B * Q := B(x Q , Kℓ(Q)) and ∆ * Q := B * Q ∩ E, and x Q is the "center" of Q as in (2.24)-(2.25).
We remark that in [HMM], the trees T were denoted by S, and were called "stopping time regimes" rather than trees.
We mention that David and Semmes,in [DS1], had previously proved a unilateral version of Lemma 3.1, in which the bilateral estimate (3.2) is replaced by the unilateral bound Next, we make a standard Whitney decomposition of Ω E := R n+1 \ E, for a given n-UR set E (in particular, Ω E is open, since n-UR sets are closed by definition). Let W = W(Ω E ) denote a collection of (closed) dyadic Whitney cubes of Ω E , so that the cubes in W form a pairwise non-overlapping covering of Ω E , which satisfy (just dyadically divide the standard Whitney cubes, as constructed in [Ste, Chapter VI], into cubes with side length 1/8 as large) and also 1 4 diam(I 1 ) ≤ diam(I 2 ) ≤ 4 diam(I 1 ) , whenever I 1 and I 2 touch.
We fix a small parameter τ 0 > 0, so that for any I ∈ W, and any τ ∈ (0, τ 0 ], the concentric dilate Moreover, for τ ≤ τ 0 small enough, and for any I, J ∈ W, we have that I * (τ) meets J * (τ) if and only if I and J have a boundary point in common, and that, if I J, then I * (τ) misses (3/4)J. Pick two parameters η ≪ 1 and K ≫ 1 (eventually, we shall take Remark 3.8. We note that W 0 Q is non-empty, provided that we choose η small enough, and K large enough, depending only on dimension and ADR, since the n-ADR condition implies that Ω E satisfies a corkscrew condition. In the sequel, we shall always assume that η and K have been so chosen.
Next, we recall a construction in [HMM,Section 3], leading up to and including in particular [HMM,Lemma 3.24]. We summarize this construction as follows.
Lemma 3.9. Let E ⊂ R n+1 be n-UR, and set Ω E := R n+1 \ E. Given positive constants η ≪ 1 and K ≫ 1, as in (3.7) and Remark 3.8, let D(E) = G ∪ B, be the corresponding bilateral Corona decomposition of Lemma 3.1. Then for each T ⊂ G, and for each Q ∈ T, the collection W 0 Q in (3.7) has an augmentation W * Q ⊂ W satisfying the following properties.
(1) W 0 Q ⊂ W * Q = W * ,+ Q ∪W * ,− Q , where (after a suitable rotation of coordinates) each I ∈ W * ,+ Q lies above the Lipschitz graph Γ T of Lemma 3.1, each I ∈ W * ,− Q lies below Γ T . Moreover, if Q ′ is a child of Q, also belonging to T, then W * ,+ Q (resp. W * ,− Q ) belongs to the same connected component of Ω E as does W * ,+ Q ′ (resp. W * ,− Q ′ ) and W * , (2) There are uniform constants c and C such that (3.10) and given T ′ , a semi-coherent subtree of T, define Then each of Ω ± T ′ is a CAD, with Chord-arc constants depending only on n, τ, η, K, and the ADR/UR constants for ∂Ω.
Remark 3.13. In particular, for each T ⊂ G, if Q ′ and Q belong to T, and if Q ′ is a dyadic child of Q, then U + Q ′ ∪ U + Q is Harnack Chain connected, and every pair of points x, y ∈ U + Q ′ ∪ U + Q may be connected by a Harnack Chain in Ω E of length at most C = C(n, τ, η, K, ADR/UR). The same is true for U − Q ′ ∪ U − Q . Remark 3.14. Let 0 < τ ≤ τ 0 /2. Given any T ⊂ G, and any semi-coherent subtree T ′ ⊂ T, define Ω ± T ′ = Ω ± T ′ (τ) as in (3.12), and similarly set Ω ± T ′ = Ω ± T ′ (2τ). Then by construction, for any , where of course the implicit constants depend on τ.
As in [HMM], it will be useful for us to extend the definition of the Whitney region U Q to the case that Q ∈ B, the "bad" collection of Lemma 3.1. Let W * Q be the augmentation of W 0 Q as constructed in Lemma 3.9, and set For Q ∈ G we shall henceforth simply write W ± Q in place of W * ,± Q . For arbitrary Q ∈ D(E), good or bad, we may then define Let us note that for Q ∈ G, the latter definition agrees with that in (3.11). Note that by construction for some uniform constants C ≥ 1 and 0 < c < 1 (see (3.4), (3.7), and (3.10)). In particular, for every Q ∈ D if follows that where we recall that D(Q) is defined in (2.26).
For future reference, we introduce dyadic sawtooth regions as follows. Given a family F of disjoint cubes {Q j } ⊂ D, we define the global discretized sawtooth relative to F by i.e., D F is the collection of all Q ∈ D that are not contained in any Q j ∈ F . We may allow F to be empty, in which case D F = D. Given some fixed cube Q, we also define the local discretized sawtooth relative to F by Note that with this convention, D(Q) = D Ø (Q) (i.e., if one takes F = Ø in (3.20)).

4.
Step 1: the set-up In the proof of Theorem 1.5, we shall employ a two-parameter induction argument, which is a refinement of the method of "extrapolation" of Carleson measures. The latter is a bootstrapping scheme for lifting the Carleson measure constant, developed by J. L. Lewis [LM], and based on the corona construction of Carleson [Car] and Carleson and Garnett [CG] (see also [HLw], [AHLT], [AHMTT], [HM1], [HM2], [HMM]). 4.1. Reduction to a dyadic setting. To set the stage for the induction procedure, let us begin by making a preliminary reduction. It will be convenient to work with a certain dyadic version of Definition 2.14. To this end, let x ∈ Ω, with δ Ω (x) < diam(∂Ω), and set ∆ x = ∆ N x = B(x, Nδ Ω (x)) ∩ ∂Ω, for some fixed N ≥ 2 as in Definition 2.11. Letx ∈ ∂Ω be a touching point for x, i.e., |x −x| = δ Ω (x). Choose x 1 on the line segment joining x tox, with δ Ω (x 1 ) = δ Ω (x)/2, and set ∆ We may therefore cover ∆ x 1 by a disjoint collection and such that the implicit constants depend only on n and ADR, and thus the cardinality M of the collection depends on n, ADR, and N. With E = ∂Ω, we make the Whitney decomposition of the set Ω E = R n+1 \ E as in Section 3 (thus, Ω ⊂ Ω E ). Moreover, for sufficiently small η and sufficiently large K in (3.7), we then have that x ∈ U Q i for each i = 1, 2, . . . , M. By hypothesis, there are constants θ 0 ∈ (0, 1], λ 0 ∈ (0, 1), and N ≥ 2 as above, such that every z ∈ Ω is a (θ 0 , λ 0 , N)-weak local John point (Definition 2.11). In particular, this is true for x 1 , hence there is a Borel set F ⊂ ∆ x 1 , with σ(F) ≥ θ 0 σ(∆ x 1 ), such that every y ∈ F may be connected to x 1 via a λ 0 -carrot path. By n-ADR, σ(∆ x 1 ) ≈ M i=1 σ(Q i ) and thus by pigeon-holing, there is at least one Q i =: Q such that σ(F ∩ Q) ≥ θ 1 σ(Q), with θ 1 depending only on θ 0 , n and ADR. Moreover, the λ 0 -carrot path connecting each y ∈ F to x 1 may be extended to a λ 1 -carrot path connecting y to x, where λ 1 depends only on λ 0 .
We have thus reduced matters to the following dyadic scenario: let Q ∈ D(∂Ω), let U Q = U Q,τ be the associated Whitney region as in (3.16), with τ ≤ τ 0 /2 fixed, and suppose that U Q meets Ω (recall that by construction U Q ⊂ Ω E = R n+1 \ E, with E = ∂Ω). For x ∈ U Q ∩ Ω, and for a constant λ ∈ (0, 1), let denote the set of y ∈ Q which may be joined to x by a λ-carrot path γ(y, x), and for θ ∈ (0, 1], set Remark 4.3. Our goal is to prove that, given λ ∈ (0, 1) and θ ∈ (0, 1], there are positive constants η and C, depending on θ, λ, and the allowable parameters, such that for each Q ∈ D(∂Ω), and for each x ∈ T Q (θ, λ), there is a Chord-arc domain Ω x , with uniformly controlled Chord-arc constants, constructed as a union ∪ k I * k of fattened Whitney boxes I * k , such that For some Q ∈ D(∂Ω), it may be that T Q is empty. On the other hand, by the preceding discussion, each x ∈ Ω belongs to T Q (θ 1 , λ 1 ) for suitable Q, θ 1 and λ 1 , so that (4.4) (with θ = θ 1 , λ = λ 1 ) implies with η 1 ≈ η, where Q is the particular Q i selected in the previous paragraph. Moreover, since x ∈ T Q ⊂ U Q , we can modify Ω x if necessary, by adjoining to it one or more fattened Whitney boxes I * with ℓ(I) ≈ ℓ(Q), to ensure that for the modified Ω x , it holds in addition that dist(x, ∂Ω x ) ℓ(Q) ≈ δ Ω (x), and therefore Ω x verifies all the conditions in Definition 2.14.
The rest of this section is therefore devoted to proving that there exists, for a given Q and for each x ∈ T Q (θ, λ), a Chord-arc domain Ω x satisfying the stated properties (when the set T Q (θ, λ) is not vacuous). To this end, we let λ ∈ (0, 1) (by Remark 4.3, any fixed λ ≤ λ 1 will suffice). We also fix positive numbers K ≫ λ −4 , and η ≤ K −4/3 ≪ λ 4 , and for these values of η and K, we make the bilateral Corona decomposition of Lemma 3.1, so that D(∂Ω) = G∪B. We also construct the Whitney collections W 0 Q in (3.7), and W * Q of Lemma 3.9 for this same choice of η and K. Given a cube Q ∈ D(∂Ω), we set  Then m is a discrete Carleson measure, i.e., recalling that D(R) is the discrete Carleson region relative to R defined in (2.26), we claim that there is a uniform constant C such that Indeed, note that for any Q ′ ∈ D, there are at most 3 cubes Q such that Q ′ ∈ D * (Q) (namely, Q ′ itself, its dyadic parent, and its dyadic grandparent), and that by n-ADR, . Thus, given any R ∈ D(∂Ω), by Lemma 3.1 part (2). Here, and throughout the remainder of this section, a generic constant C, and implicit constants, are allowed to depend upon the choice of the parameters η and K that we have fixed, along with the usual allowable parameters. With (4.8) in hand, we therefore have

4.2.
Induction Hypothesis and Outline of Proof. As mentioned above, our proof will be based on a two parameter induction scheme. Given λ ∈ (0, λ 1 ] fixed as above, we recall that the set F car (x, Q, λ) is defined in (4.1). The induction hypothesis, which we formulate for any a ≥ 0, and any θ ∈ (0, 1] is as follows:

H[a, θ]
There is a positive constant c a = c a (θ) < 1 such that for any given Q ∈ D(∂Ω), if (4.10) m(D(Q)) ≤ aσ(Q), and if there is a subset V Q ⊂ U Q ∩ Ω for which there is a Chord-arc domain Ω i Q which is the interior of the union of a collection of fattened Whitney cubes I * , and whose Chord-arc constants depend only on dimension, λ, a, θ, and the ADR constants for , where the sum runs over those i such that U i Q meets V * Q .

Some geometric observations
We begin with some preliminary observations. In what follows we have fixed λ ∈ (0, λ 1 ] and two positive numbers K ≫ λ −4 , and η ≤ K −4/3 ≪ λ 4 , for which the bilateral Corona decomposition of D(∂Ω) in Lemma 3.1 is applied. We now fix k 0 ∈ N, k 0 ≥ 4, such that Suppose that there are points x ∈ U Q ∩ Ω and y ∈ Q ′ , that are connected by a λ-carrot path γ = γ(y, x) in Ω. Then γ meets U Q ′ ∩ Ω.
We shall also require the following. We recall that by Lemma 3.9, for Q ∈ T ⊂ G, the Whitney region U Q has the splitting lying above (resp., below) the Lipschitz graph Γ T of Lemma 3.1.
Lemma 5.3. Let Q ′ ⊂ Q, and suppose that Q ′ and Q both belong to G, and moreover that both Q ′ and Q belong to the same tree T ⊂ G. Suppose that y ∈ Q ′ and x ∈ U Q ∩ Ω are connected via a λ-carrot path γ(y, x) in Ω, and assume that there is a point z ∈ γ(y, x) ∩ U Q ′ ∩ Ω (by Lemma 5.2 we know that such a z exists provided Proof. We suppose for the sake of contradiction that, e.g., x ∈ U + Q , and that z ∈ U − Q ′ . Thus, in traveling from y to z and then to x along the path γ(y, x), one must cross the Lipschitz graph Γ T at least once between z and x. Let y 1 be the first point on γ(y, x) ∩ Γ T that one encounters after z, when traveling toward x. By Lemma 3.9, , as in Lemma 3.1. On the other hand, where in the last step we have used Lemma 3.9. This contradicts our choice of η ≪ λ 4 .
We now form a chain of consecutive dyadic cubes where the introduced notation P i ⊳ P i+1 means that P i is the dyadic child of P i+1 , that is, P i ⊂ P i+1 and ℓ(P i+1 ) = 2ℓ(P i ). Let P := P i 0 , 1 ≤ i 0 ≤ M + 1, be the smallest of the cubes P i such that y 1 ∈ B * P i . Setting P ′ := P i 0 −1 , we then have that y 1 ∈ B * P , and y 1 B * P ′ . By the coherency of T, it follows that P ∈ T, so by (3.2), (5.4) δ Ω (y 1 ) ≤ ηℓ(P) .
Lemma 5.6. Fix λ ∈ (0, 1). Given Q ∈ D(∂Ω) and a non-empty set V Q ⊂ U Q ∩ Ω, such that each x ∈ V Q may be connected by a λ-carrot path to some y ∈ Q, set where we recall that F car (x, Q, λ) is the set of y ∈ Q that are connected via a λ-carrot path to x (see (4.1)).
is also a λ-carrot path, for the same constant λ. All the conclusions in the lemma follow easily from the construction by letting V Q ′ = y∈F Q ∩Q ′ y ′ (y).
Remark 5.8. It follows easily from the previous proof that under the same assumptions, if one further assumes that ℓ(Q ′ ) < 2 −k 0 ℓ(Q), we can then repeat the argument with both Q ′ and (Q ′ ) * (the dyadic parent of Q ′ ) to obtain respectively V Q ′ and V (Q ′ ) * . Moreover, this can be done in such a way that every point in V Q ′ (resp. V (Q ′ ) * ) belongs to a λ-carrot path which also meets V (Q ′ ) * (resp. V Q ′ ), connecting U Q and Q ′ .
Given a family F := {Q j } ⊂ D(∂Ω) of pairwise disjoint cubes, we recall that the "discrete sawtooth" D F is the collection of all cubes in D(∂Ω) that are not contained in any Q j ∈ F (see (3.19)), and we define the restriction of m (cf. (4.6), (4.7)) to the sawtooth D F by We then set Let us note that we may allow F to be empty, in which case D F = D and m F is simply m. We note that the following claim, and others in the sequel, remain true when F is empty; sometimes trivially so, and sometimes with some straightforward changes that are left to the interested reader.
Claim 5.10. Given Q ∈ D(∂Ω), and a family F = , each Q j ∈ F , and every dyadic child Q ′ j of any Q j ∈ F , belong to the good collection G, and moreover, every such cube belongs to the same tree T ⊂ G. In particular, the collection of cubes which are the maximal elements of the trees T in G), then by construction α Q ′ = σ(Q ′ ) for that cube (see (4.6)), so by definition of m and m F , we would have a contradiction. Similarly, if some Q j ∈ F (respectively, Q ′ j ∈ F ′ ) were in M ∪ B, then its dyadic parent (respectively, dyadic grandparent) Q * j would belong to D F ∩ D(Q), and by definition α Q * j = σ(Q * j ), so again we reach a contradiction. Consequently, F ∪ F ′ ∪ (D F ∩ D(Q)) does not meet M ∪ B, and the claim follows.

Construction of chord-arc subdomains
For future reference, we now prove the following. Recall that for Q ∈ G, U Q has precisely two connected components U ± Q in R n+1 \ ∂Ω.
, and a non-empty subcollection F * ⊂ F , such that:

is a sufficiently large positive constant; and
(iii) F * has a disjoint decomposition F * = F * + ∪ F * − , where for each Q j ∈ F * ± , there is a Chord-arc subdomain Ω ± Q j ⊂ Ω, consisting of a union of fattened , and with uniform control of the Chord-arc constants.
Define a semi-coherent subtree T * ⊂ T by and for each choice of ± for which F * ± is non-empty, set Then for κ large enough, depending only on allowable parameters, Ω ± Q is a Chordarc domain, with chord arc constants depending only on the uniformly controlled Chord-arc constants of Ω ± Q j and on the other allowable parameters. Moreover, ∩ Ω, and Ω ± Q is a union of fattened Whitney cubes. Remark 6.3. Note that we define Ω ± Q if and only if F * ± is non-empty. It may be that one of F * + , F * − is empty, but F * + and F * − cannot both be empty, since F * is non-empty by assumption.
Proof of Lemma 6.1. Without loss of generality we may assume that Ω Q j ± is not contained in Ω ± T * for all Q j ∈ F * (otherwise we can drop those cubes from F * ). On the other hand, we notice that Ω ± Q is a union of (open) fattened Whitney cubes (assuming that it is non-empty): each Ω ± Q j has this property by assumption, as does Ω ± T * by construction. We next observe that if Ω + Q (resp. Ω − Q ) is non-empty, then it is contained in Ω. Indeed, by construction, Ω + Q is non-empty if and only if F * + is non-empty. In turn, F * + is non-empty if and only if there is some Q j ∈ F * such that U + Q j ⊂ Ω + Q j ⊂ Ω, and moreover, the latter is true for every Q j ∈ F * + , by definition. But each such Q j belongs to T * , hence U + Q j ⊂ Ω + T * , again by construction (see (3.12)). Thus, Ω + T * meets Ω, and since Ω + T * ⊂ R n+1 \ ∂Ω, therefore Ω + T * ⊂ Ω. Combining these observations, we see that Ω + Q ⊂ Ω. Of course, the same reasoning applies to Ω − Q , provided it is non-empty.
In addition, since T * ⊂ T, and since K ≫ K 1/2 , by Lemma 3.9 we have Ω ± It therefore remains to establish the Chord-arc properties. It is straightforward to prove the interior corkscrew condition and the upper n-ADR bound, and we omit the details. Thus, we must verify the Harnack Chain condition, the lower n-ADR bound, and the exterior corkscrew condition.
6.1. Harnack Chains. Suppose, without loss of generality, that Ω + Q is non-empty, and let x, y ∈ Ω + Q , with |x − y| = r. If x and y both lie in Ω + T * , or in the same Ω + Q j , then we can connect x and y by a suitable Harnack path, since each of these domains is Chord-arc. Thus, we may suppose either that 1) x ∈ Ω + T * and y lies in some Ω + Q j , or that 2) x and y lie in two distinct Ω + Q j 1 and Ω + Q j 2 . We may reduce the latter case to the former case: by the separation property (ii) in Lemma 6.1, we must have r κ max diam(Ω + ) , so given case 1), we can connect x ∈ Ω + Q j 1 to the center z 1 of some I * 1 ⊂ U + Q 1 , and y ∈ Ω + Q j 2 to the center z 2 of some Finally, we can connect z 1 and z 2 using that Ω + T * is Chord-arc. Hence, we need only construct a suitable Harnack Chain in Case 1). We note that by assumption and construction, U where c ′ ≤ 1 is a sufficiently small positive constant to be chosen. Since y ∈ Ω + Q j ⊂ B * Q j , we then have that x ∈ 2B * Q j , so by the construction of Ω + T * and the separation property (ii), it follows that δ Ω (x) ≥ cℓ(Q j ), where c is a uniform constant depending only on the allowable parameters (in particular, this fact is true for all x ∈ Ω + T * ∩2B * Q j , so it does not depend on the choice of c ′ < 1). Now choosing c ′ ≤ c/2 (eventually, it may be even smaller), we find that Since Ω + Q j and Ω + T * are each the interior of a union of fattened Whitney cubes, it follows that there are Whitney cubes I and J, with x ∈ I * , y ∈ J * , and where the implicit constants depend on K. For c ′ small enough in (6.4), depending on the implicit constants in the last display, and on the parameter τ in (3.5), this can happen only if I * and J * overlap (recall that we have fixed τ small enough that I * and J * overlap if and only if I and J have a boundary point in common), in which case we may trivially connect x and y by a suitable Harnack Chain.
On the other hand, suppose that we may find such a z, since U + Q j is a union of fattened Whitney cubes, all of length ℓ(I * ) ≈ ℓ(Q j ); just take z to be the center of such an I * ). We may then construct an appropriate Harnack Chain from y to x by connecting y to z via a Harnack Chain in the Chord-arc domain Ω + Q j , and z to x via a Harnack Chain in the Chord-arc domain Ω + T * .
6.2. Lower n-ADR and exterior corkscrews. We will establish these two properties essentially simultaneously. Again suppose that, e.g., Ω + Q is non-empty. Let Our main goal at this stage is to prove the following: , with c a uniform positive constant depending only upon allowable parameters (including κ). Indeed, momentarily taking this estimate for granted, we may combine (6.5) with the interior corkscrew condition to deduce the lower n-ADR bound via the relative isoperimetric inequality [EG,p. 190]. In turn, with both the lower and upper n-ADR bounds in hand, (6.5) implies the existence of exterior corkscrews (see, e.g., [HM2,Lemma 5.7]).
Thus, it is enough to prove (6.5). We consider the following cases.
Case 2: B(x, r/2) meets ∂Ω + Q j for at least one Q j ∈ F * + , and r ≤ κ 1/2 ℓ(Q j 0 ), where Q j 0 is chosen to have the largest length ℓ(Q j 0 ) among those Q j such that ∂Ω + Q j meets B(x, r/2). We now further split the present case into subcases.
M is a large number to be chosen. Then B(z, (Mκ 1/2 ) −1 r) ⊂ B(x, r), for M large enough. In addition, we claim that B(z, (Mκ 1/2 ) −1 r) misses Ω + T * ∪ ∪ j j 0 Ω + Q j . The fact that B(z, (Mκ 1/2 ) −1 r) misses every other Ω + Q j , j j 0 , follows immediately from the restriction r ≤ κ 1/2 ℓ(Q j 0 ), and the separation property (ii). To see that , by the construction of Ω + T * and the separation property (ii). Thus, the claim follows, for a sufficiently large (fixed) choice of M. Since B(z, (Mκ 1/2 ) −1 r) misses Ω + T * and all other Ω + Q j , we inherit an exterior corkscrew point in the present case (depending on M and κ) from the Chord-arc domain Ω + Q j 0 . Again (6.5) follows.
Otherwise, by the separation property (ii), the remaining possibility in the present scenario is that , the claim follows.
On the other hand, since x ∈ ∂Ω + Q , there is a J ∈ W with ℓ(J) ≈ ℓ(Q j 0 ), such that J * is not contained in Ω + Q . We then have an exterior corkscrew point in J * ∩ B(x, r), and (6.5) follows in this case.
Case 3: B(x, r/2) meets ∂Ω + Q j for at least one Q j ∈ F * + , and r > κ 1/2 ℓ(Q j 0 ), where as above Q j 0 has the largest length ℓ(Q j 0 ) among those Q j such that ∂Ω + Q j meets B(x, r/2). In particular then, r ≫ 2Kℓ( by our choice of k 1 . By this fact, and the definition of Ω T * , we have ) . Using then that Ω ± T * is connected, we see that a path within Ω ± T * joining U ± Q j with U ± Q must meet ∂B(x, 3Kℓ(Q j )). Hence we can find y ± ∈ Ω ± T * ∩ ∂B(x, 3Kℓ(Q j )). By Lemma 3.9, Ω + T * and Ω − T * are disjoint (they live respectively above and below the graph Γ T ), so a path joining y + and y − within ∂B(x, 3Kℓ(Q j )) meets some With the point x 1 in hand, we note that By the exterior corkscrew condition for Ω + T * , , for some constant c 1 depending only on n and the ADR/UR constants for ∂Ω, by Lemma 3.9. Also, for each Ω + Q j whose boundary meets B(x 1 , r/4) \ Ω + T * (and thus meets B(x, r/2)), , for all such Q j . We now make the following claim.
Observe that by the second containment in (6.6), we obtain (6.5) as an immediate consequence of (6.10), and thus the proof will be complete once we have established Claim 6.9.
Proof of Claim 6.9. To prove the claim, we suppose first that where the sum runs over those j such that B * Q j meets B(x 1 , r/4) \ Ω + T * , and c 1 is the constant in (6.7). In that case, (6.10) holds with c 2 = c 1 /2 (and even with B(x 1 , r/4)), by definition of Ω + Q (see (6.2)), and the fact that On the other hand, if (6.11) fails, then summing over the same subset of indices j, we have We now make a second claim: Claim 6.13. For j appearing in the previous sum, we have (6.14) Taking the latter claim for granted momentarily, we insert estimate (6.14) into (6.12) and sum, to obtain By the separation property (ii), the balls κ 1/4 B * Q j are pairwise disjoint, and by assumption Ω + Q j ⊂ B * Q j . Thus, for any given Moreover, as noted above (see (6.8) and the ensuing comment), κ 1/4 B * Q j ⊂ B(x 1 , r/2) for each j under consideration in (6.11)-(6.15). Claim 6.9 now follows.
Proof of Claim 6.13. There are two cases: In the latter case, by the exterior corkscrew condition for Q j , and (6.14) follows, finishing the proof of Claim 6.13.
Next, (6.6) and (6.10) yield (6.5) in the present case and hence the proof of Lemma 6.1 is complete.
We shall deduce H[M 0 , 1] (see Section 4.2) from the following pair of claims.  .7)-(4.9) above. Fix Q ∈ D(E). Let a ≥ 0 and b > 0, and suppose that m D(Q) ≤ (a + b) σ(Q). Then there is a family F = {Q j } ⊂ D(Q) of pairwise disjoint cubes, and a constant C depending only on n and the ADR constant such that We refer the reader to [HM2,Lemma 7.2] for the proof. We remark that the lemma is stated in [HM2] in the case that E is the boundary of a connected domain, but the proof actually requires only that E have a dyadic cube structure, and that σ be a nonnegative, dyadically doubling Borel measure on E. In our case, we shall of course apply the lemma with E = ∂Ω, where Ω is open, but not necessarily connected.
In the present scenario θ = 1, that is, σ(F Q ) = σ(Q) (see (4.11) and (5.7)), which implies σ(F Q ∩ Q ′ ) = σ(Q ′ ). We apply Lemma 5.6 to obtain V Q ′ ⊂ U Q ′ ∩ Ω and the corresponding F Q ′ which satisfies σ(F Q ′ ) = σ(Q ′ ). That is, (4.11) holds for Q ′ , with θ = 1. Consequently, we may apply the induction hypothesis to By Lemma 5.6, and since k 1 > k 0 , each y ∈ V * Q ′ lies on a λ-carrot path connecting some y ∈ Q ′ to some x ∈ V Q ; let V * * Q denote the set of all such x, and let U * * Q (respectively, U * Q ′ ) denote the collection of connected components of U Q (resp., of may be joined to some corresponding component in U * * Q , via one of the carrot paths. After possible renumbering, we designate this component as respectively, that are joined by this carrot path, and we let γ i be the portion of the carrot path joining x i to y i (if there is more than one such path or component, we just pick one). We also let V * Q = {x i } i be the collection of all of the selected points x i . We let W i be the collection of Whitney cubes meeting γ i , and we then define By the definition of a λ-carrot path, since ℓ(Q ′ ) ≈ k 1 ℓ(Q), and since Ω i Q ′ is a CAD, one may readily verify that Ω i Q is also a CAD consisting of a union ∪ k I * k of fattened Whitney cubes I * k . We omit the details. Moreover, by construction, that the analogue of (7.6) holds with Q ′ replaced by Q, and with c a replaced by c k 1 c a .
It remains to verify that . By the induction hypothesis, and our choice of We therefore need only to consider I * with I ∈ W i . For such an I, by definition there is a point z i ∈ I ∩γ i and y i ∈ Q ′ , so that z i ∈ γ(y i , x i ) and thus, where in the last inequality we have used (3.17) and the fact that x i ∈ U Q . Hence, for every z ∈ I * by (3.4) by our choice of the parameters K and λ.
We then obtain the conclusion of H[a + b, 1] in the present case.
In this case, we apply Lemma 7.3 to obtain a pairwise disjoint family F = {Q j } ⊂ D(Q) such that (7.4) and (7.5) hold. In particular, by our choice of b = 1/(2C), so that the conclusions of Claim 5.10 hold. We set Then by (7.5) where ρ ∈ (0, 1) is defined by We claim that Indeed, were this not true for some Q j , then by definition of F good and pigeon-holing . This contradicts the assumptions of the current case.
Note also that Q F good by (7.12) and Q F bad by (7.5), hence F ⊂ D(Q) \ {Q}. By (7.7) and Claim 5.10, there is some tree T ⊂ G so that T ′′ = (D F ∪ F ∪ F ′ ) ∩ D(Q) is a semi-coherent subtree of T, where F ′ denotes the collection of all dyadic children of cubes in F .
. In this case, Q has an ample overlap with the boundary of a Chord-arc domain with controlled Chord-arc constants. Indeed, let T ′ = D F ∩ D(Q) which, by (7.7) and Claim 5.10, is a semi-coherent subtree of some T ⊂ G. Hence, by Lemma 3.9, each of Ω ± T ′ is a CAD with constants depending on the allowable parameters, formed by the union of fattened Whitney boxes, which satisfies Ω ± T ′ ⊂ B * Q ∩ Ω (see (3.11), (3.12), and (3.18)). Moreover, by [HMM,Proposition A.14] and [HM2,Proposition 6.3] and our current assumptions, Recall that in establishing H[a + b, 1], we assume that there is a set V Q ⊂ U Q ∩ Ω for which (4.11) holds with θ = 1. Pick then x ∈ V Q and set V * For the sake of specificity assume that x ∈ U + Q ∩ Ω hence, in particular, U + Q ⊂ Ω + T ′ ⊂ Ω. Note also that U + Q is the only component of U Q meeting V * Q . All these together give at once that the conclusion of H[a + b, 1] holds in the present case.
Case 2b: σ(F 0 ) < 1 2 ρσ(Q). In this case by (7.10) In addition, by the definition of F good (7.9), and pigeon-holing, every Q j ∈ F good has a dyadic child Q ′ j (there could be more children satisfying this, but we just pick one) so that . We apply Lemma 5.6 (recall (7.12)) to obtain . That is, (4.11) holds for Q ′ j , with θ = 1. Consequently, recalling that Q ′ j ∈ T ⊂ G (see Claim 5.10), and applying the induction hypothesis to Q formed by a union of fattened Whitney By a covering lemma argument, for a sufficiently large constant κ ≫ K 4 , we may extract a subcollection F * good ⊂ F good so that {κB * Q j } Q j ∈F * good is a pairwise disjoint family, and In particular, by (7.13), where the implicit constants depend on ADR, K, and the dilation factor κ. By the induction hypothesis, and by construction (7.15) and n-ADR, where Ω Q j is equal either to Ω + Q j or to Ω − Q j (if (7.17) holds for both choices, we arbitrarily set Ω Q j = Ω + Q j ). Combining (7.17) with (7.16), we obtain (7.18) We now assign each Q j ∈ F * good either to F * + or to F * − , depending on whether we chose Ω Q j satisfying (7.17) to be Ω + Q j , or Ω − Q j . We note that at least one of the sub-collections F * ± is non-empty, since for each j, there was at least one choice of "+' or "-" such that (7.17) holds for the corresponding choice of Ω Q j . Moreover, the two collections are disjoint, since we have arbitrarily designated Ω Q j = Ω + Q j in the case that there were two choices for a particular Q j .
To proceed, as in Lemma 6.1 we set which is semi-coherent by construction. For F * ± non-empty, we now define (7. 19) Observe that by the induction hypothesis, and our construction (see (7.15) and the ensuing comment), for an appropriate choice of ±, U ± Q j ⊂ Ω Q j ⊂ B * Q j , and since ℓ(Q j ) ≤ 2 −k 1 ℓ(Q), by (7.18) and Lemma 6.1, with F * = F * good , each (non-empty) choice of Ω ± Q defines a Chord-arc domain with the requisite properties. Thus, we have proved Claim 7.2 and therefore, as noted above, it follows that H[M 0 , 1] holds.

Step 3: bootstrapping θ
In this last step, we shall prove that there is a uniform constant ζ ∈ (0, 1) such that for each θ ∈ ] holds for any given θ 1 ∈ (0, 1]. As noted above, it then follows that Theorem 1.5 holds, as desired.
In turn, it will be enough to verify the following.
Proof of Claim 8.1. The proof will be a refinement of that of Claim 7.2. We are given some θ ∈ (0, 1] such that H[M 0 , θ] holds, and we assume that H[a, (1 − ϑ)θ] holds, for some a ∈ [0, M 0 ) and ϑ ∈ (0, 1). Set b = 1/(2C), where as before C is the constant in (7.4). Consider a cube Q ∈ D(∂Ω) with m(D(Q)) ≤ (a + b)σ(Q). Suppose that there is a set V Q ⊂ U Q ∩ Ω such that (4.11) holds with θ replaced by (1 − ϑβ)θ, for some β ∈ (0, 1) to be determined. Our goal is to show that for a sufficiently small, but uniform choice of β, we may deduce the conclusion of the induction hypothesis, with C a+b , c a+b in place of C a , c a .
By assumption, and recalling the definition of F Q in (5.7), we have that (4.11) holds with constant (1 − ϑβ)θ, i.e., As in the proof of Claim 7.2, we fix k 1 > k 0 (see (5.1)) large enough so that 2 k 1 > 100K. There are two principal cases. The first is as follows.
We split Case 1 into two subcases.
In this case, we follow the Case 1 argument for θ = 1 in Section 7 mutatis mutandis, so we merely sketch the proof. By Lemma 5.6, we may construct V Q ′ and F Q ′ so that F Q ∩ Q ′ = F Q ′ and hence σ(F Q ′ ) ≥ (1 − ϑ)θσ(Q ′ ). We may then apply the induction hypothesis H[a, (1 − ϑ)θ] in Q ′ , and then proceed exactly as in Case 1 in Section 7 to construct a subset V * Q ⊂ V Q and a family of Chord-arc domains Ω i Q satisfying the various desired properties, and such that The conclusion of H[a + b, (1 − ϑβ)θ] then holds in the present scenario.
In the scenario of Case 1b, this leads to that is, Note that we have the dyadic doubling estimate where M 1 = M 1 (k 1 , n, ADR). Combining this estimate with (8.3), we obtain We now choose β ≤ 1/(M 1 +1), so that (1−β)/M 1 ≥ β, and therefore the expression in square brackets is at least 1. Consequently, by pigeon-holing, there exists a particular where the latter is defined as in (5.7), with Q ′′ 0 in place of Q. By assumption, H[M 0 , θ] holds, so combining (8.4) with the fact that (4.10) holds with a = M 0 for every Q ∈ D(∂Ω), we find that there exists a subset V * } i enjoying all the appropriate properties relative to Q ′′ 0 . Using that ℓ(Q ′′ 0 ) ≈ k 1 ℓ(Q), we may now proceed exactly as in Case 1a above, and also Case 1 in Section 7, to construct V * Q and {Ω i Q } i such that the conclusion of H[a + b, (1 − ϑβ)θ] holds in the present case also.
Recall that F Q is defined in (5.7), and satisfies (8.2). We define F 0 = Q \ ( F Q j ) as in (7.8), and F good := F \ F bad as in (7.9). Let G 0 := F good Q j . Then as above (see (7.10)), where again ρ = ρ(M 0 , b) ∈ (0, 1) is defined as in (7.11). Just as in Case 2 for θ = 1 in Section 7, we have that (see (7.12)). Hence, the conclusions of Claim 5.10 hold. We first observe that if σ(F 0 ) ≥ εσ(Q), for some ε > 0 to be chosen (depending on allowable parameters), then the desired conclusion holds. Indeed, in this case, we may proceed exactly as in the analogous scenario in Case 2a in Section 7: the promised Chord-arc domain is again simply one of Ω ± T ′ , since at least one of these contains a point in V Q and hence in particular is a subdomain of Ω. The constant c a+b in our conclusion will depend on ε, but in the end this will be harmless, since ε will be chosen to depend only on allowable parameters.
We may therefore suppose that Next, we refine the decomposition F = F good ∪ F bad . With ρ as in (7.11) and (8.5), we choose β < ρ/4. Set , and define F (2) bad := F bad \ F (1) bad . We split the remaining part of Case 2 into two subcases. The first of these will be easy, based on our previous arguments.
. We may then apply H[M 0 , θ] in Q ′ , and proceed exactly as we did in Case 1b above with the cube Q ′′ 0 , which enjoyed precisely the same properties as does our current Q ′ . Thus, we draw the desired conclusion in the present case.
The main case is the following.
Case 2b: Every Q j ∈ F (1) bad satisfies ℓ(Q j ) ≤ 2 −k 1 ℓ(Q). Observe that by definition, good . For future reference, we shall derive a certain ampleness estimate for the cubes in F * . By (8.2), where in the last step have used (8.7) and (8.8). Observe that Using (8.10) and (8.11), for ε ≪ 4ρ −1 − 1 ϑβθ, we obtain and thus We now make the following claim.
With Claim 8.13 in hand, let us return to the proof of Case 2b of Claim 8.1. We begin by noting that by definition of F (1) bad , and Lemma 5.6, we can apply H[M 0 , θ] to any Q j ∈ F (1) bad , hence for each such Q j there is a family of Chord-arc domains {Ω i Q j } i satisfying the desired properties. Now consider Q j ∈ F (1) good . Since F (1) good ⊂ F good , by pigeon-holing Q j has a dyadic child Q ′ j satisfying (8.18) m D(Q ′ j ) ≤ aσ(Q ′ j ) , (there may be more than one such child, but we just pick one). Our immediate goal is to find a child Q ′′ j of Q j , which may or may not equal Q ′ j , for which we may construct a family of Chord-arc domains {Ω i Q ′′ j } i satisfying the desired properties. To this end, we assume first that Q ′ j satisfies . In this case, we set Q ′′ j := Q ′ j , and using Lemma 5.6, by the induction hypothesis H[a, (1 − ϑ)θ], we obtain the desired family of Chord-arc domains.
We therefore consider the case . In this case, we shall select Q ′′ j Q ′ j . Recall that we use the notation Q ′′ ⊳ Q to mean that Q ′′ is a dyadic child of Q. Set where M 1 = M 1 (n, ADR). We also note that By (8.20), it follows that In turn, using (8.22), we obtain By the dyadic doubling estimate (8.21), this leads to Choosing β ≤ ρ/(4(M 1 + 1)), we find that the expression in square brackets is at least 1, and therefore, by pigeon holing, we can pick Q ′′ j ∈ F ′′ j satisfying (8.23) σ(F Q ∩ Q ′′ j ) ≥ θσ(Q ′′ j ) . Hence, using Lemma 5.6, we see that the induction hypothesis H[M 0 , θ] holds for Q ′′ j ∈ F ′′ j , and once again we obtain the desired family of Chord-arc domains. Recall that we have constructed our packing measure m in such a way that each Q j ∈ F , as well as all of its children, along with the cubes in D F ∩ D(Q), belong to the same tree T; see Claim 5.10. This means in particular that for each such Q j , the Whitney region U Q j has exactly two components U ± Q j ⊂ Ω ± T , and the analogous statement is true for each child of Q j . This fact has the following consequences: Remark 8.24. For each Q j ∈ F (1) bad , and for the selected child Q ′′ j of each Q j ∈ F (1) good , the conclusion of the induction hypothesis produces at most two Chord-arc domains with i = 1 corresponding "+", and i = 2 corresponding to "-", respectively.
By the induction hypothesis, for each Q j ∈ F (1) bad ∪ F (1) good (and by n-ADR, in the case of F (1) good ), the Chord-arc domains Ω i Q j that we have constructed satisfy where the sum has either one or two terms, and where the implicit constant depends either on M 0 and θ, or on a and (1 − ϑ)θ, depending on which part of the induction hypothesis we have used. In particular, for each such Q j , there is at least one choice of index i such that Ω i Q j =: Ω Q j satisfies (if the latter is true for both choices i = 1, 2, we arbitrarily choose i = 1, which we recall corresponds to "+"). Combining the latter bound with Claim 8.13, and recalling that ε has now been fixed depending only on allowable parameters, we see that . By a covering lemma argument, we may extract a subfamily F * ⊂ F (1) bad ∪ F (1) good such that {κB * Q j } Q j ∈F * is pairwise disjoint, where again κ ≫ K 4 is a large dilation factor, and such that Let us now build (at most two) Chord-arc domains Ω i Q satisfying the desired properties. Recall that for each Q j ∈ F * , we defined the corresponding Chord-arc domain Ω Q j := Ω i Q j , where the choice of index i (if there was a choice), was made so that (8.26) holds. We then assign each Q j ∈ F * either to F * + or to F * − , depending on whether we chose Ω Q j satisfying (8.26) to be Ω 1 Q j = Ω + Q j , or Ω 2 Q j = Ω − Q j . We note that at least one of the sub-collections F * ± is non-empty, since for each j, there was at least one choice of index i such that (8.26) holds with Ω Q j := Ω i Q j . Moreover, the two collections are disjoint, since we have arbitrarily designated Ω Q j = Ω 1 Q j (corresponding to "+") in the case that there were two choices for a particular Q j . We further note that if Q j ∈ F * ± , then Ω Q j = Ω ± Q j ⊃ U ± Q j . We are now in position to apply Lemma 6.1. Set which is a semi-coherent subtree of T, with maximal cube Q. Without loss of generality, we may suppose that F * + is non-empty, and we then define and similarly with "+" replaced by "-", provided that F * − is also non-empty. Observe that by the induction hypothesis, and our construction (see Remarks 8.24 and 8.25, and Lemma 3.9), for an appropriate choice of "±", U ± Q j ⊂ Ω Q j ⊂ B * Q j , and since ℓ(Q j ) ≤ 2 −k 1 ℓ(Q), by (8.27) and Lemma 6.1, each (non-empty) choice defines a Chord-arc domain with the requisite properties. This completes the proof of Case 2b of Claim 8.1 and hence that of Theorem 1.5.
Part 2: Proof of Theorem 1.6 9. Preliminaries for the Proof of Theorem 1.6 9.1. Uniform rectifiability. Recall the definition of n-uniform rectifiable (n-UR) sets in Definition 2.3. Given a ball B ⊂ R n+1 , we denote where the infimum is taken over all the affine n-planes that intersect B.
The constant 3 multiplying B Q in the estimate above can be replaced by any number larger than 1. For the proof, see [DS2,.
Theorem 9.3. Let Ω ⊂ R n+1 , n ≥ 1, be an open set satisfying an interior corkscrew condition, with n-ADR boundary, such that the harmonic measure in Ω belongs to weak-A ∞ . Then ∂Ω is n-UR.
9.2. Harmonic measure. From now on we assume that Ω ⊂ R n+1 is an open set with n-ADR boundary such that the harmonic measure in Ω belongs to weak-A ∞ . We denote by σ the surface measure in ∂Ω, that is, σ = H n ⌊ ∂Ω . We also consider the dyadic lattice D associated with σ as in Lemma 2.23. The AD-regularity constant of ∂Ω is denoted by C 0 . We denote by ω p the harmonic measure with pole at p of Ω, and by g(·, ·) the Green function. Much as before we write δ Ω (x) = dist(x, ∂Ω).
The following is also well known.
We remark that the previous lemma is also valid in the case n > 1 without the n-ADR assumption. In the case n = 1 this holds under the 1-ADR assumption, and also in the more general situation where Ω satisfies the CDC. This follows easily from [AH, Lemmas 3.4 and 3.5]. Notice that n-ADR implies the CDC in R n+1 (for any n), by standard arguments.
The following lemma is also known. See [HLMN, Lemma 3.14], for example.
where C and α depend on n and the AD-regularity of ∂Ω. In particular, The next result provides a partial converse to Lemma 9.6.
Hence, for a small enough, we derive ω p (2Q) k 0 ℓ(P) n−1 y∈3B P :δ Ω (y)≥aℓ(P) g(p, y) dy, which implies the existence of the point q required in the lemma.
9.3. Harnack chains and carrots. It will be more convenient for us to work with Harnack chains instead of curves. The existence of a carrot curve is equivalent to having what we call a good chain between points.
Let x ∈ Ω, y ∈ Ω be such that δ Ω (y) ≤ δ Ω (x), and let C > 1. A C-good chain (or C-good Harnack chain) from x to y is a sequence of balls B 1 , B 2 , . . . (finite or infinite) contained in Ω such that x ∈ B 1 and either where N is the number of elements of the sequence if this is finite, and moreover the following hold: Abusing language, sometimes we will omit the constant C and we will just say "good chain" or "good Harnack chain".
Observe that in the definitions of carrot curves and good chains, the order of x and y is important: having a carrot curve from x to y is not equivalent to having one from y to x, and similarly with good chains.
Lemma 9.10. There is a carrot curve from x ∈ Ω to y ∈ Ω if and only if there is a good Harnack chain from x to y.
Proof. Let γ be a carrot curve from x to y. We can assume y ∈ Ω, since if y ∈ ∂Ω, we can obtain this case by taking a limit of points y j ∈ Ω converging to y. Let {B j } N j=1 be a Vitali subcovering of the family {B(z, δ Ω (z)/10) : z ∈ γ} and let r B j stand for the radius and x B j for the center of B j . So the balls B j are disjoint and 5B j cover γ. Note that for t > 0, if t < r B j ≤ 2t, In particular, since the B j 's are disjoint, by volume considerations, there can only be boundedly many B j of radius between t/2 and t, say. Moreover, we may order the balls B j so that x ∈ 5B 1 and B j+1 is a ball B k such that 5B k ∩ 5B j ∅ and 5B k contains the point from γ ∩ h:5B h ∩5B j ∅ 5B h which is maximal in the natural order induced by γ (so that x is the minimal point in γ). Then for j > i, This implies 5B 1 , 5B 2 , . . . is a C-good chain for a sufficiently big C. Now suppose that we can find a good chain from x to y, call it B 1 , . . . , B N . Let γ be the path obtained by connecting their centers in order. Let z ∈ γ. Then there is a j such that z ∈ [x B j , x B j+1 ], the segment joining x B j and x B j+1 . Since {B i } i is a good chain, We would like to note that the implicit constants do not depend on N. Indeed, from the properties of the good chain it easily follows that Thus, γ is a carrot curve from x to y.
10. The Main Lemma for the proof of Theorem 1.6 Because of the absence of doubling conditions on harmonic measure under the weak-A ∞ assumption, to prove Theorem 1.6 we cannot use arguments similar to the ones in [AH] or [Azz]. Instead, we prove a local result which involves only one pole and one ball which has its own interest. This is the Main Lemma 10.2 below.
Let B ⊂ R n+1 be a ball centered at ∂Ω and let p ∈ Ω. We restate Definition 2.20 in the following form: ω p satisfies the weak-A ∞ condition in B if for every ε 0 ∈ (0, 1) there exists δ 0 ∈ (0, 1) such that the following holds: for any subset E ⊂ B ∩ ∂Ω, In the next sections we will prove the following.

Main Lemma 10.2.
Let Ω ⊂ R n+1 have n-uniformly rectifiable boundary. Let R 0 ∈ D and let p ∈ Ω \ 4B R 0 be a point such that and ω p (R 0 ) ≥ c ′ > 0. Suppose that ω p satisfies the weak-A ∞ condition in B R 0 . Then there exists a subset Con(R 0 ) ⊂ R 0 and a constant c ′′ > 0 with σ(Con(R 0 )) ≥ c ′′ σ(R 0 ) such that each point x ∈ Con(R 0 ) can be joined to p by a carrot curve. The constant c ′′ and the constants involved in the carrot condition only depend on c, c ′ , n, the weak-A ∞ condition, and the n-UR character of ∂Ω.
The notation Con(·) stands for "connectable". It is easy to check that Theorem 1.6 follows from this result. First notice that the assumptions of the theorem imply that ∂Ω is n-uniformly rectifiable by Theorem 9.3. Consider now any x ∈ Ω and take a point ξ ∈ ∂Ω such that |x − ξ| = δ Ω (x). Then we consider the point p in the segment [x, ξ] such that |p − ξ| = 1 16 δ Ω (x). By Lemma 9.5, we have ω p (B(ξ, 1 8 δ Ω (x))) 1, because p ∈ 1 2 B(ξ, 1 8 δ Ω (x)). Hence, by covering B(ξ, 1 8 δ Ω (x)) ∩ Ω with cubes R ∈ D contained in B(ξ, 1 4 δ Ω (x)) ∩ ∂Ω with side length comparable to δ Ω (x) we deduce that at least one these cubes, call it R 0 , satisfies ω p (R 0 ) 1. Further, by taking the side length small enough, we may also assume that p 4B R 0 . Since ω p satisfies the weak-A ∞ property in B R 0 (by the assumptions in Theorem 1.6), we can apply the Main Lemma 10.2 above and infer that there exists a subset F := Con(R 0 ) ⊂ R 0 with σ(F) ≥ c ′ σ(R 0 ) δ Ω (x) n such that all y ∈ F can be joined to x by a carrot curve, which proves that Ω satisfies the weak local John condition and concludes the proof of Theorem 1.6.
Two essential ingredients of the proof of the Main Lemma 10.2 are a corona type decomposition (whose existence is ensured by the n-uniform rectifiability of the boundary) and the Alt-Caffarelli-Friedman monotonicity formula [ACF]. This formula is used in some of the connectivity arguments below. This allows to connect by carrot curves corkscrew points where the Green function is not too small to other corkscrew points at a larger distance from the boundary where the Green function is still not too small (see Lemma 11.11 for the precise statement). The use of the Alt-Caffarelli-Friedman formula is not new to problems involving harmonic measure and connectivity (see, for example, [AGMT]). However, the way it is applied here is new, as far as we know.
Two important steps of the proof of the Main Lemma 10.2 (and so of Theorem 1.6) are the Geometric Lemma 14.5 and the Key Lemma 15.2. An essential idea consists of distinguishing cubes with "two well separated big corkscrews" (see Section 13.4 for the precise definition). In the Geometric Lemma 10.2 we construct two disjoint open sets satisfying a John condition associated to trees involving this type of cubes, so that the boundaries of the open sets are located in places where the Green function is very small. This construction is only possible because the associated tree involves only cubes with two well separated big corkscrews. The existence of these cubes is an obstacle for the construction of carrot curves. However, in a sense, in the Key Lemma 15.2 we take advantage of their existence to obtain some delicate estimates for the Green function on some corkscrew points.
We claim now that to prove he Main Lemma 10.2 we can assume that Ω = R n+1 \ ∂Ω. To check this, let Ω, p, and R 0 satisfy the assumptions in the Main Lemma. Consider the open set V = R n+1 \ ∂Ω. Then the harmonic measure ω p in Ω coincides with the harmonic measure ω p V in V (the fact that V is not connected does not disturb us). Thus V, p, and R 0 satisfy the assumptions in the Main Lemma, and moreover V = R n+1 \ ∂Ω = R n+1 \ ∂V. Assuming the Main Lemma to be valid in this particular case, we deduce that there exists a subset Con(R 0 ) ⊂ R 0 and a constant c ′′ > 0 with σ(Con(R 0 )) ≥ c ′′ σ(R 0 ) such that each point x ∈ Con(R 0 ) can be joined to p by a carrot curve in V. Now just observe that if γ is one of this carrot curves and it joints p and x ∈ Con(R 0 ) ⊂ ∂V = ∂Ω, then γ is contained in V except for its end-point x. By connectivity, since p ∈ Ω ∩ γ, γ must be contained in Ω, except for the end-point x. Hence, γ is a carrot curve with respect to Ω.
Sections 11-16 are devoted to the proof of Main Lemma 10.2. To this end, we will assume that Ω = R n+1 \ ∂Ω.
11. The Alt-Caffarelli-Friedman formula and the existence of short paths 11.1. The Alt-Caffarelli-Friedman formula. Recall the following well known result of Alt-Caffarelli-Friedman (see [CS,Theorems 12.1 and 12.3
In the case of equality we have the following result (see [PSU,Theorem 2.9]).
Theorem 11.5. Let B(x, R) and u 1 , u 2 be as in Theorem 11.1. Suppose that J(x, r a ) = J(x, r b ) for some 0 < r a < r b < R. Then either one or the other of the following holds: there exists a unit vector e and constants k 1 , k 2 > 0 such that We will also need the following auxiliary lemma.
For any ε > 0, there exists some δ > 0 such that if with J(·, ·) defined in (11.2), then either one or the other of the following holds: there exists a unit vector e and constants k 1 , k 2 > 0 such that The constant δ depends only on n, α, C 1 , ε.
Proof. Suppose that the conclusion of the lemma fails. Then, by replacing u i (y) by 1 R u i (Ry + x), we can assume that x = 0 and R = 1. Let ε > 0, and for each δ = 1/k and i = 1, 2, consider functions u i,k satisfying the assumptions of the lemma and such that neither (a) nor (b) holds for them. By Lemma 11.6, there exist subsequences (which we still denote by {u i,k } k ) which converge uniformly in B(0, 3 2 ) and weakly in W 1,2 (B(0, 3 2 )) to some functions u i ∈ W 1,2 (B(0, 3 2 )) ∩ C(B(0, 3 2 )), and moreover, |∇u i (y)| 2 |y| n−1 dy both for r = 1 and r = 1/2. Clearly, the functions u i are non-negative, subharmonic, and u 1 · u 2 = 0. Hence, by Theorem 11.5, one of the following holds: there exists a unit vector e and constants k 1 , k 2 > 0 such that However, the fact that neither (a) nor (b) holds for any pair u 1,k , u 2,k , together with the uniform convergence of {u i,k } k , implies that neither (a') nor (b') can hold, and thus we get a contradiction.
Proof. This follows easily from Lemmas 9.6 and 9.9.
Proof. All the parameters in the lemma will be fixed along the proof. We assume that A 1 ≫ κ −1 > 1. First note that we may assume that r < 2A −1 1 |x 0 − p|. Otherwise, we just take a point q ′ ∈ Ω such that |p − q ′ | = δ Ω (p)/2, which clearly satisfies the properties in (11.13). Further, both q and q ′ belong to the open connected set for a sufficiently small c 2 > 0. The fact that U is connected is well known. This follows from the fact that, for any λ > 0, any connected component of {g(p, ·) > λ} should contain p. Otherwise there would be a connected component where g(p, ·)−λ is positive and harmonic with zero boundary values. So, by maximum principle, g(p, ·) − λ should equal λ in the whole component, which is a contradiction. So there is only one connected component.
We just let γ be a curve contained in U. Note that for a sufficiently small a > 0 because, by boundary Hölder continuity, if dist(x, ∂Ω) ≤ δ Ω (p)/2. Further, the fact that g(p, x) ≤ c|x − p| 1−n ensures that U ⊂ B(p, Cδ Ω (p)), for a sufficiently big constant C depending on r/r 0 .
So from now on we assume that r < 2A −1 1 |x 0 − p|. By Lemma 11.10 we know there exists some point q ∈ Ω such that (11.14) δ Ω (p) n , with c depending on κ and Λ.
Assume that q and q cannot be joined by a curve γ as in the statement of the lemma. Otherwise, we choose q ′ = Q and we are done. For t > 0, consider the open We fix t > 0 small enough such that q, q ∈ V 2t ⊂ V t . Such t exists by (11.12) and (11.14), and it may depend on Λ, λ, r/r 0 .
Let V 1 and V 2 be the respective components of V t to which q and q belong. We have V 1 ∩ V 2 = ∅, because otherwise there is a curve contained in V t ⊂ B(x 0 , 1 4 A 1 r) which connects q and q, and further this is far away from ∂Ω. Indeed, we claim that (11.15) dist(V t , ∂Ω) A 1 ,Λ,t,r/r 0 r 0 .
To see this, note that by the Hölder continuity of g(p, ·) in B(x 0 , 1 2 A 1 r), for all x ∈ V t , we have where in the last inequality we used Lemma 9.7 and that x 0 ∈ WA(p, Λ). This yields our claim.
Next we wish to apply the Alt-Caffarelli-Friedman formula with It is clear that both satisfy the hypotheses of Theorem 11.1. For i = 1, 2 and 0 < s < A 1 r, we denote The condition (i) follows from (11.4) and the fact that (11.16) g(p, y) s δ Ω (p) n for all y ∈ B(x 0 , s), which holds by Lemma 9.7 and subharmonicity, since x 0 ∈ WA(p, Λ). Concerning (ii), note first that |∇u 1 (y)| δ Ω (p) n g (p, y) δ Ω (y) τ 0 δ Ω (p) n r 0 δ Ω (p) n = 1 for all y ∈ B(q, τ 0 r 0 /2), where we first used Cauchy estimates and then the pointwise bounds of g(·, ·) in (11.16) with s ≈ δ Ω (y). Thus, using also that q ∈ V 2t , we infer that u 1 (y) > t r 0 /2 in some ball B(q, ctr 0 ) with c possibly depending on Λ, λ, r/r 0 . Analogously, we deduce that u 2 (y) > t r 0 /2 in some ball B( q, ctr 0 ). Let B be the largest open ball centered at q not intersecting ∂V 1 and let y 0 ∈ ∂V 1 ∩ ∂B. Then, by considering the convex hull H ⊂ B of B(q, ctr 0 ) and y 0 and integrating in spherical coordinates (with the origin in y 0 ), one can check that H |∇u 1 | dy t r n+1 0 .
An analogous estimate holds for u 2 , and then it easily follows that which implies (ii). We leave the details for the reader. From the conditions (i) and (ii) and the fact that J(x, r) is non-decreasing we infer that J(x 0 , s) ≈ Λ,λ,r/r 0 1 for 2r < s < 1 4 A 1 r. and also (11.17) J i (x 0 , s) ≈ Λ,λ,r/r 0 1 for i = 1, 2 and 2r < s < 1 4 A 1 r.
Thus B(q ′ , 2 h−2 r) ⊂ Ω and so q ′ is at a distance at least 2 h−2 r from ∂Ω, and also Further, since q and q ′ are both in V 1 by definition, there is a curve γ which joins q and q ′ contained in V 1 satisfying dist(γ, ∂Ω) A 1 ,Λ,t,r/r 0 r 0 , by (11.15). So q ′ satisfies all the required properties in the lemma and we are done.

Types of cubes
From now on we fix R 0 ∈ D and p ∈ Ω and we assume that we are under the assumptions of the Main Lemma 10.2.
We need now to define two families HD and LD of high density and low density cubes, respectively. Let A ≫ 1 be some fixed constant. We denote by HD (high density) the family of maximal cubes Q ∈ D which are contained in R 0 and satisfy We also denote by LD (low density) the family of maximal cubes Q ∈ D which are contained in R 0 and satisfy Lemma 12.1. We have Proof. By Vitali's covering theorem, there exists a subfamily I ⊂ HD so that the cubes 2Q, Q ∈ I, are pairwise disjoint and Then, since σ is doubling, we obtain Next we turn our attention to the low density cubes. Since the cubes from LD are pairwise disjoint, we have From the above estimates and the fact that the harmonic measure belongs to weak-A ∞ (cf. (10.1)), we infer that if A is chosen big enough, then and thus As a consequence, denoting G 0 = R 0 \ (B H ∪ B L )), we deduce that again using the fact that ω p belongs to weak-A ∞ in B R 0 . So we have: Lemma 12.2. Assuming A big enough, the set G 0 : with the implicit constants depending on C 0 and the weak-A ∞ condition in B R 0 .
We denote by G the family of those cubes Q ∈ D(R 0 ) which are not contained in P∈HD∪LD P. In particular, such cubes Q ∈ G do not belong to HD ∪ LD and From this fact, it follows easily that G 0 is contained in the set WA(p, Λ) defined in Section 11.2, assuming Λ big enough, and so Lemma 12.2 ensures that (11.9) holds.
The following lemma is an immediate consequence of Lemma 11.10.
Lemma 12.4. For every cube Q ∈ G there exists some point z Q ∈ 2B Q ∩ Ω such that δ Ω (z Q ) ≥ κ 0 ℓ(Q) and for some κ 0 , c 3 > 0, which depend on A and on the weak-A ∞ constants in B R 0 .
If z Q ∈ 2B Q ∩ Ω and δ Ω (z Q ) ≥ κ 0 ℓ(Q), we say that z Q is κ 0 -corkscrew for Q. If (12.5) holds, we say that z Q is a c 3 -good corkscrew for Q. Abusing notation, quite often we will not write "for Q".
We will need the following auxiliary result: Lemma 12.6. Let Q ∈ D and let z Q be a λ-good c 4 -corkscrew, for some λ, c 4 > 0. Suppose that ℓ(Q) ≥ c 5 ℓ(R 0 ). Then there exists some C-good Harnack chain that joins z Q and p, with C depending on λ, c 5 .
From the fact that g(p, x) ≤ |p − x| 1−n for all x ∈ Ω, we infer that any x ∈ U satisfies λ ℓ(Q) σ(R 0 ) < g(p, x) ≤ 1 |p − x| n−1 . Therefore, So U ⊂ B(p, C 2 ℓ(R 0 )) for some C 2 depending on λ and c 5 . Next we consider a Besicovitch covering of γ with balls B i of radius c 6 ℓ(R 0 )/2. By volume considerations, it easily follows that the number of balls B i is bounded above by some constant C 3 depending on λ and c 5 , and thus this is a C-good Harnack chain, with C = C(λ, c 5 ).
Lemma 12.7. There exists some constant κ 1 with 0 < κ 1 ≤ κ 0 such that the following holds for all λ > 0. Let Q ∈ G, Q R 0 , and let z Q be a λ-good κ 1 -corkscrew. Then there exists some cube R ∈ G with Q R ⊂ R 0 and ℓ(R) ≤ C ℓ(Q) and a λ ′ -good κ 1 -corkscrew z R such that z Q and z R can be joined by a C ′ (λ)-good Harnack chain, with λ ′ > 0 and C depending on λ.
The proof below yields a constant λ ′ < λ. On the other hand, the lemma ensures that z R is still a κ 1 -corkscrew, which will be important for the arguments to come.
Proof. This follows easily from Lemma 11.11. For completeness we will show the details.
By choosing Λ = Λ(A) > 0 big enough, G 0 ∩ Q ⊂ WA(p, Λ) and thus there exists some x 0 ∈ Q ∩ WA(p, Λ). We let where κ 0 is defined in Lemma 12.4 and κ in Lemma 11.10 (and thus it depends only on A and C 0 ). We apply Lemma 11.11 to x 0 , q = z Q , with r 0 = 3r(B Q ), λ 0 ≈ λ, and r = 4r(B Q ). To this end, note that Hence there exists q ′ ∈ B(x 0 , A 1 r) such that and such that q and q ′ can be joined by a curve γ such that (12.9) γ ⊂ {y ∈ B(x 0 , A 1 r) : dist(y, ∂Ω) > a 1 r 0 }, with λ 1 , A 1 , a 1 depending on C 0 , A, λ, κ 1 . Now let R ∈ D be the cube containing x 0 such that Observe that Also, we may assume that ℓ(R) ≤ ℓ(R 0 ) because otherwise we have ℓ(Q) A 1 δ Ω (p) and then the statement in the lemma follows from Lemma 12.6. So we have Q R ⊂ R 0 . From (12.8) we get and g(p, q ′ ) ≥ c λ 1 2κ ℓ(R) σ(R 0 ) .
Hence, q ′ is a λ ′ -good κ 1 -corkscrew, for λ ′ = cλ 1 2κ. From (12.9) and arguing as in the end of the proof of Lemma 12.6 we infer that z Q = q and z R = q ′ can be joined by a C(λ)-good Harnack chain.
From now on we will assume that all corkscrew points for cubes Q ∈ G are κ 1corkscrews, unless otherwise stated.
13. The corona decomposition and the Key Lemma 13.1. The corona decomposition. Recall that the bβ coefficient of a ball was defined in (9.1). For each Q ∈ D, we denote bβ(Q) = bβ ∂Ω (100B Q ). Now we fix a constant 0 < ε ≪ min(1, κ 1 ). Given R ∈ D(R 0 ), we denote by Stop(R) the maximal family of cubes Q ∈ D(R) \ {R} satisfying that either Q G or bβ Q > ε, where Q is the parent of Q. Recall that the family G was defined in (12.3). Note that, by maximality, Stop(R) is a family of pairwise disjoint cubes.
We define In particular, note that Stop(R) T(R).
We now define the family of the top cubes with respect to R 0 as follows: first we define the families Top k for k ≥ 1 inductively. We set Assuming that Top k has been defined, we set and then we define Top k .
Notice that the family of cubes Q ∈ D(R 0 ) with ℓ(Q) ≤ 2 −10 ℓ(R 0 ) which are not contained in any cube P ∈ HD ∪ LD is contained in R∈Top T(R), and this union is disjoint. Also, all the cubes in that union belong to G.
The following lemma is an easy consequence of our construction. Its proof is left for the reader.
Lemma 13.1. We have Remark that the last inequality holds for any cube Q ∈ Stop(R) because its parent Q belongs to T(R) and so Q is not contained in any cube from HD, which implies that ω p (2Q) ≤ ω p (2 Q) A σ( Q) σ(R 0 ) ≈ A σ(Q) σ(R 0 ) . Using that ∂Ω is n-UR (by the assumption in the Main Lemma 10.2), it is easy to prove that the cubes from Top satisfy a Carleson packing condition. This is shown in the next lemma.
Lemma 13.2. We have

Then we get
Note now that, because of the stopping conditions, for all Q ∈ Top, if P ∈ Stop(Q) ∩ G, then the parent P of P satisfies bβ ∂Ω (100B P ) > ε. Hence, by Theorems 9.2 and 9.3, On the other hand, the cubes P ∈ Stop(Q) \ G with Q ∈ Top do not contain any cube from Top, by construction. Hence, they are disjoint and thus

By an analogous reason,
Using (13.3) and the estimates above, the lemma follows.
Given a constant K ≫ 1, next we define By Chebyshev and the preceding lemma, we have Therefore, if K is chosen big enough (depending on M(ε) and the constants on the weak-A ∞ condition), by Lemma 12.2 we get and thus We distinguish now two types of cubes from Top. We denote by Top a the family of cubes R ∈ Top such that T(R) = {R}, and we set Top b = Top \ Top a . Notice that, by construction, if R ∈ Top b , then bβ(R) ≤ ε. On the other hand, this estimate may fail if R ∈ Top a . 13.2. The truncated corona decomposition. For technical reasons, we need now to define a truncated version of the previous corona decomposition. We fix a big natural number N ≫ 1. Then we let Top (N) be the family of the cubes from Top with side length larger than 2 −N ℓ(R 0 ). Given R ∈ Top (N) we let T (N) (R) be the subfamily of the cubes from T(R) with side length larger than 2 −N ℓ(R 0 ), and we let Stop (N) (R) be a maximal subfamily from Stop(R) ∪ D N (R 0 ), where D N (R 0 ) is the subfamily of the cubes from D(R 0 ) with side length 2 −N ℓ(R 0 ). We also denote Top (N) 13.3. The Key Lemma. The main ingredient for the proof of the Main Lemma 10.2 is the following result.
This lemma will be proved in the next Sections 14 and 15. Using this result, in Section 16 we will build the required carrot curves for the Main Lemma 10.2, which join the pole p to points from a suitable big piece of R 0 . If the reader prefers to see how this is applied before its long proof, he may go directly to Section 16. A crucial point in the Key Lemma is that the constant ε in the definition of the stopping cubes of the corona decomposition does not depend on the constants λ or η above.
To prove the Key Lemma 13.5 we will need first to introduce the notion of "cubes with well separated big corkscrews" and we will split T (N) (R) into subtrees by introducing an additional stopping condition involving this type of cubes. Later on, in Section 14 we will prove the "Geometric Lemma", which relies on a geometric construction which plays a fundamental role in the proof of the Key Lemma.
13.4. The cubes with well separated big corkscrews. Let Q ∈ D be a cube such that bβ(Q) ≤ C 4 ε. For example, Q might be a cube from Q ∈ T (N) (R) ∪ Stop (N) (R), with R ∈ Top (N) b (which in particular implies that bβ(R) ≤ ε). We denote by L Q a best approximating n-plane for bβ(Q), and we choose z 1 Q and z 2 Q to be two fixed points in B Q such that dist(z i Q , L Q ) = r(B Q )/2 and lie in different components of R n+1 \ L Q . So z 1 Q and z 2 Q are corkscrews for Q. We will call them "big corkscrews". Since any corkscrew x for Q satisfies δ Ω (x) ≥ κ 1 ℓ(Q) and we have chosen ε ≪ κ 1 , it turns out that dist(x, L Q ) ≥ 1 2 κ 1 ℓ(Q) ≫ ε ℓ(Q).
As a consequence, x can be joined either to z 1 Q or to z 2 Q by a C-good Harnack chain, with C depending only on n, C 0 , κ 1 , and thus only on n, C 0 and the weak-A ∞ constants in B R 0 . The following lemma follows by the same reasoning: Lemma 13.6. Let Q, Q ′ ∈ D be cubes such that bβ(Q), bβ(Q ′ ) ≤ C 4 ε and Q ′ is the parent of Q. Let z i Q , z i Q ′ , for i = 1, 2, be big corkscrews for Q and Q ′ respectively. Then, after relabeling the corkscrews if necessary, z i Q can be joined to z i Q ′ by a Cgood Harnack chain, with C depending only on n, C 0 , κ 1 .
Given Γ > 0, we will write Q ∈ WSBC(Γ) (or just Q ∈ WSBC, which stands for "well separated big corkscrews") if bβ(Q) ≤ C 4 ε and the big corkscrews z 1 Q , z 2 Q can not be joined by any Γ-good Harnack chain. The parameter Γ will be chosen below. For the moment, let us say that Γ −1 ≪ ε. The reader should think that in spite of bβ(Q) ≤ C 4 ε, the possible existence of "holes of size C εℓ(Q) in ∂Ω" makes possible the connection of the big corkscrews by means of Γ-Harnack chains passing through these holes. Note that if Q WSBC(Γ), then any pair of corkscrews for Q can be connected by a C(Γ)-good Harnack chain, since any of these corkscrews can be joined by a good chain to one of the big corkscrews for Q, as mentioned above.
13.5. The tree of cubes of type WSBC and the subtrees. Given R ∈ Top (N) b , denote by Stop WSBC (R) the maximal subfamily of cubes from Q ∈ D(R) which satisfy that either Also, denote by T WSBC (R) the cubes from D(R) which are not strictly contained in any cube from Stop WSBC (R). So this tree is empty if R WSBC(Γ).
For each Q ∈ Stop WSBC (R) \ Stop(R), we denote and the union is disjoint. Observe also that we have the partition 14. The geometric lemma 14.1. The geometric lemma for the tree of cubes of type WSBC. Let R ∈ Top (N) b and suppose that T WSBC (R) ∅. We need now to define a family End(R) of cubes from D, which in a sense can be considered as a regularized version of Stop(R). The first step consists of introducing the following auxiliary function: Observe that d R is 1-Lipschitz.
For each x ∈ ∂Ω we take the largest cube Q x ∈ D such that x ∈ Q x and We consider the collection of the different cubes Q x , x ∈ ∂Ω, and we denote it by End(R).
Proof. The proof is a routine task. For the reader's convenience we show the details. To show (a), consider x ∈ 50B P . Since d R (·) is 1-Lipschitz and, by definition, d R (x P ) ≥ 300 ℓ(P), we have To prove the converse inequality, by the definition of End(R), there exists some z ′ ∈ P, the parent of P, such that d R (z ′ ) ≤ 300 ℓ( P) = 600 ℓ(P).

Also, we have
The statement (b) is an immediate consequence of (a), and (c) follows easily from (b). To show (d), observe that, for any S ∈ T WSBC (R), Thus, In particular, choosing S = R, we deduce and thus, using again that dist(P, R) ≤ 20ℓ(R), it follows that P ⊂ 22R. Let S 0 ∈ T WSBC (R) be such that d R (x P ) = ℓ(S 0 ) + dist(x P , S 0 ), and let Q ∈ D be the smallest cube such that S 0 ⊂ Q and P ⊂ 22Q. Since S 0 ⊂ R and P ⊂ 22R, we deduce that S 0 ⊂ Q ⊂ R, implying that Q ∈ T WSBC (R). So it just remains to check that ℓ(Q) ≤ 2000 ℓ(P). To this end, consider a cube Q ⊃ S 0 such that ℓ(P) + ℓ(S 0 ) + dist(P, S 0 ) ≤ ℓ( Q) ≤ 2 ℓ(P) + ℓ(S 0 ) + dist(P, S 0 ) .
From the first inequality, it is clear that P ⊂ 2 Q and then, by the definition of Q, we infer that Q ⊂ Q. This inclusion and the second inequality above imply that By (a) we know that d R (x P ) ≤ 900 ℓ(P), and so we derive ℓ(Q) ≤ 2000 ℓ(P).
As in Section 3, we make a standard Whitney decomposition of the open set Ω. With a harmless abuse of notation we let W = W(Ω) denote a collection of (closed) dyadic Whitney cubes of Ω, so that the cubes in W form a pairwise non-overlapping covering of Ω, which satisfy for some M 0 > 20 and D 0 ≥ 1 (i) 10I ⊂ Ω; (ii) M 0 I ∩ ∂Ω ∅; (iii) there are at most D 0 cubes I ′ ∈ W such that 10I ∩ 10I ′ ∅. Further, for such cubes I ′ , we have ℓ(I ′ ) ≈ ℓ(I), where ℓ(I ′ ) stands for the side length of I ′ .
To construct this Whitney decomposition one just needs to replace each cube I ∈ W, as in [Ste, Chapter VI], by its descendants I ′ ∈ D k (I), for some fixed k ≥ 1. For each I ∈ W, as much as in Lemma 9.6, we denote by B I a ball concentric with I and radius C 5 ℓ(I), where C 5 is a universal constant big enough so that and whenever p 5I. Obviously, the ball B I intersects ∂Ω, and the family {B I } I∈W does not have finite overlapping. Given a bounded measurable set F ⊂ R n+1 with |F| > 0, and a function f ∈ L 1 loc (R n+1 ), we denote by m F f the mean of f in F with respect to Lebesgue measure. That is, To state the Geometric Lemma we need some additional notation. Given a cube R ′ ∈ T WSBC (R), we denote by T WSBC (R ′ ) the family of cubes from D with side length at most ℓ(R ′ ) which are contained in 100B R ′ and are not contained in any cube from End(R). We also denote by End(R ′ ) the subfamily of the cubes from End(R) which are contained in some cube from T WSBC (R ′ ). Note that T WSBC (R ′ ) is not a tree, in general, but a union of trees.
In order to define the open sets V 1 , V 2 described in the lemma, first we need to associate some open sets U 1 (Q), U 2 (Q) to each Q ∈ T WSBC (R ′ ) ∪ End(R ′ ). We distinguish two cases: • For Q ∈ T WSBC (R ′ ), we let J i (Q) be the family of Whitney cubes I ∈ W which intersect {y ∈ 20B Q : dist(y, L Q ) > ε 1/4 ℓ(Q)} and are contained in the same connected component of R n+1 \ L Q as z i Q , and then we set 1.1 int(I).
• For Q ∈ End(R ′ ) the definition of U i (Q) is more elaborated. First we consider an auxiliary ball B Q , concentric with B Q , such that 19B Q ⊂ B Q ⊂ 20B Q and having thin boundaries for ω p . This means that, for some absolute constant C, The existence of such ball B Q follows by well known arguments (see for example [To,p.370]). Next we denote by J(Q) the family of Whitney cubes I ∈ W which intersect B Q and satisfy ℓ(I) ≥ θ ℓ(Q) for θ ∈ (0, 1) depending on γ (the reader should think that θ ≪ ε and that θ = 2 − j 1 for some j 1 ≫ 1), and we set 1.1 int(I). 4 To guarantee the existence of the sets V i and the fact that they are contained in Ω we use the assumption that Ω = R n+1 \ ∂Ω.
For a fixed i = 1 or 2, let {D i j (Q)} j≥0 be the connected components of U(Q) which satisfy one of the following properties: either z i Q ∈ D i j (Q) (recall that z i Q is a big corkscrew for Q), or there exists some y ∈ D i j (Q) such that g(p, y) > γ ℓ(Q) σ(R 0 ) −1 and there is a C 6 (γ, θ)-good Harnack chain that joins y to z i Q , for some constant C 6 (γ, θ) to be chosen below. Then we let U i (Q) = j D i j (Q). After reordering the sequence, we assume that z i Q ∈ D i 0 (Q).
In the case Q ∈ T WSBC (R ′ ), from the definitions, it is clear that the sets U i (Q) are open and connected and (14.9) In the case Q ∈ End(R ′ ), the sets U i (Q) may fail to be connected. However, (14.9) still holds if Γ is chosen big enough (which will be the case). Indeed, if some component D i j can be joined by C 6 (γ, θ)-good Harnack chains both to z 1 Q and z 2 Q , then there is a C(γ, θ)-good Harnack chain that joins z 1 Q to z 2 Q , and thus Q does not belong to WSBC(c 6 Γ) if Γ is taken big enough, which cannot happen by Lemma 14.3. Note also that the two components of , because bβ(Q) ≤ Cε and we assume θ ≪ ε. The following is immediate: Lemma 14.10. Assume that we relabel appropriately the sets U i (P) and corkscrews z i P for P ∈ T WSBC (R ′ ) ∪ End(R ′ ). Then for all Q, Q ∈ T WSBC (R ′ ) ∪ End(R ′ ) such that Q is the parent of Q we have where c depends at most on n and C 0 .
The labeling above can be chosen inductively. First we fix the sets U i (T ) and corkscrews x i T for every maximal cube T from T WSBC (R ′ ) (contained in 100B R ′ and with side length equal to ℓ(R ′ )). Further we assume that, for any maximal cube T , the corkscrew x i T is at the same side of L R ′ as z i R ′ , for each i = 1, 2 (this property will be used below). Later we label the sons of each T so that (14.11) holds for any son Q of T . Then we proceed with the grandsons of T , and so on. We leave the details for the reader.
The following result will be used later to prove the property (e)(i).
Proof. By the definition of U i (Q), it suffices to show that y belongs to some component D i j (Q) and that there is a C 6 (γ, θ)-good Harnack chain that joins y to z i Q . To this end, observe that by the boundary Hölder continuity of g(p, ·), where in the last inequality we used Lemma 9.7. Thus, and if θ is small enough, then y belongs to some connected component of the set U(Q) in (14.8). By Lemma 14.2(d) there is a cube Q ′ ∈ T WSBC (R) such that Q ⊂ 22Q ′ and ℓ(Q ′ ) ≈ ℓ(Q). In particular, WA(p, Λ) ∩ Q ′ ⊃ G 0 ∩ Q ′ ∅ and thus, by applying Lemma 11.11 with q = y and r 0 = Cr(B Q ) (for a suitable C > 1), it follows that there exists a κ 1 -corkscrew y ′ ∈ C(γ) B Q , with C(γ) > 20 say, such that y can be joined to y ′ by a C ′ (γ)-good Harnack chain. Assuming that the constant k 0 (γ) in Lemma 14.5 is big enough, it turns out that y ′ ∈ CB Q ′′ for some Q ′′ ∈ T WSBC (R) such that 22Q ′′ ⊃ Q. Since all the cubes S such that Q ⊂ S ⊂ 22Q ′′ satisfy bβ(S ) ≤ C ε, by applying Lemma 13.6 repeatedly, it follows that y ′ can be joined either to z 1 Q or z 2 Q by a C ′′ (γ)-good Harnack chain. Then, joining both Harnack chains, it follows that y can be joined either to z 1 Q or z 2 Q by a C ′′′ (γ)-good Harnack chain. So y belongs to one of the components D i j , assuming C 6 (γ, θ) big enough.
From now on we assume θ small enough and C 6 (γ, θ) big enough so that the preceding lemma holds. Also, we assume θ ≪ ε 4 . We define Next we will show that V 1 ∩ V 2 = ∅. Since the number of cubes Q ∈ T WSBC (R ′ ) ∪ End(R ′ ) is finite (because of the truncation in the corona decomposition), this is a consequence of the following: Lemma 14.13. Suppose Γ is big enough in the definition of WSBC (depending on θ). For all P, Q ∈ T WSBC (R ′ ) ∪ End(R ′ ), we have Proof. We suppose that ℓ(Q) ≥ ℓ(P) We also assume that U 1 (P) ∩ U 2 (Q) ∅ and then we will get a contradiction. Notice first that if ℓ(P) = ℓ(Q) = 2 − j ℓ(R ′ ) for some j ≥ 0, then the corkscrews z i P and z i Q are at the same side of L Q for each i = 1, 2. This follows easily by induction on j.
Let P be the ancestor of P such that ℓ( P) = ℓ(Q). From the fact that U 1 (P) ∩ U 2 (Q) ∅, we deduce that 20B P ∩ 20B Q ∅ and thus 20B P ∩ 20B Q ∅, and so 20B P ⊂ 60B Q . This implies that z 1 P is in the same connected component as z 1 Q and also that dist([z 1 Q , z 1 P ], ∂Ω) ℓ(Q), because bβ(100B Q ) ≤ ε ≪ 1 and they are at the same side of L Q .
Case 2. Suppose now that Q ∈ End(R ′ ). The arguments are quite similar to the ones above. In this case, the cubes from J 2 (Q) have side length at least θ ℓ(Q) and thus at least one of the cubes from J 1 (P) has side length at least c θ ℓ(Q), which implies that ℓ(P) ≥ c ′ θ ℓ(Q).
Again construct a curve γ ′′ = γ ′′ (z 1 Q , z 2 Q ) which joins z 1 Q to z 2 Q by gathering [z 1 Q , z 1 P ], γ ′ (z 1 P , z 1 P ), and γ(z 1 P , z 2 Q ). This is contained in CB Q (for some C > 1 possibly depending on γ) and satisfies dist(γ ′′ , ∂Ω) ≥ c θ 2 ℓ(Q). From this fact we deduce that z 1 Q and z 2 Q can be joined by C(θ)-good Harnack chain. Taking Γ big enough (depending on C(θ)), this implies that the big corkscrews for Q can be joined by a (c 6 Γ)-good Harnack chain, which contradicts Lemma 14.3.
By the definition of V 1 and V 2 it is clear that the properties (a), (b) and (c) in Lemma 14.5 hold. So to complete the proof of the lemma it just remains to prove (d) and (e).
Proof of Lemma 14.5(d). Let x ∈ (∂V 1 ∪ ∂V 2 ) ∩ 10B R ′ . We have to show that there exists some S ∈ End(R ′ ) such that x ∈ 2B S . To this end we consider y ∈ ∂Ω such that |x − y| = δ Ω (x). Since x R ′ ∈ ∂Ω, it follows that y ∈ 20B R ′ . Let S ∈ End(R ′ ) be such that y ∈ S . Observe that (14.14) We claim that x ∈ 2B S . Indeed, if x 2B S , taking also into account (14.14), there exists some ancestor Q of S contained in 100B R ′ such that x ∈ 2B Q and δ Ω (x) = |x − y| ≈ ℓ(Q). From the fact that S Q ⊂ 100B R ′ we deduce that Q ∈ T WSBC (R ′ ). By the construction of the sets U i (Q), it is immediate to check that the condition that δ Ω (x) ≈ ℓ(Q) implies that x ∈ U 1 (Q) ∪ U 2 (Q). Thus x ∈ V 1 ∪ V 2 and so x ∂(V 1 ∪ V 2 ) = ∂V 1 ∪ ∂V 2 (for this identity we use that dist(V 1 , V 2 ) > 0), which is a contradiction.
To show (e), first we need to prove the next result: Lemma 14.15. For each i = 1, 2, we have Proof. Clearly, we have So it suffices to show that (14.16) From the definition of U i (P), it follows easily that (14.17) δ Ω (x) ε 1/4 ℓ(P).
Proof of Lemma 14.5(e). Let P ∈ End(R ′ ) be such that 2B P ∩ 10B R ′ ∅. The statement (i) is an immediate consequence of Lemma 14.12. In fact, this lemma implies that any y ∈ 2B P such that g(p, y) > γ ℓ(P) σ(R 0 ) −1 is contained in U 1 (P) ∪ U 2 (P) and thus in V 1 ∪ V 2 . In particular, y ∂(V 1 ∪ V 2 ) = ∂V 1 ∪ ∂V 2 . Thus, if y ∈ 2B P ∩ ∂V i , then It is easy to check that this implies the statement (i) in Lemma 14.5(e) (possibly after replacing γ by Cγ).
Next we turn our attention to (ii). To this end, denote by J P the subfamily of the cubes Q ∈ End(R ′ ) such that 30B Q ∩ 2B P ∅. By Lemma 14.15, We will show that (14.20) I∈W P ℓ(I) n ℓ(P) n and where W P the family of Whitney cubes I ⊂ V 1 ∪V 2 such that 1.1I∩∂(V 1 ∪V 2 )∩2B P ∅. To this end, observe that, by (14.19) and the construction of U i (Q), for each I ∈ W P there exists some Q ∈ J P such that I ⊂ 30B Q and either ℓ(I) = θℓ(Q) or 1.1I ∩ ∂ B Q ∅. Using the n-ADRity of σ, it is immediate to check that for each Q ∈ J P , I⊂30B Q : ℓ(I)=θℓ(Q) ℓ(I) n ℓ(Q) n .

Also,
Since the number of cubes Q ∈ J P is uniformly bounded (by Lemma 14.2(b)) and ℓ(Q) ≈ ℓ(P), the above inequalities yield the first estimate in (14.20).
To prove the second one we also distinguish among the two types of cubes I ∈ J P above. First, by the bounded overlap of the balls B I such that ℓ(I) = θ ℓ(Q), we get (14.21) since the balls B I in the sum are contained CB P for a suitable universal constant C > 1. To deal with the cubes I ∈ W such that 1.1I ∩ ∂ B Q ∅ we intend to use the thin boundary property of B Q in (14.7). To this end, we write I∈W: where U d (A) stands for the d-neighborhood of A. By (14.7) it follows that ω p (U 2 −k ℓ(Q) (∂ B Q )) 2 −k ω p (C ′ B Q ), and thus I∈W: for a suitable C > 1. Together with (14.21), this yields the second inequality in (14.20), which completes the proof of Lemma 14.5(e).

Proof of the Key Lemma
We fix R 0 ∈ D and a corkscrew point p ∈ Ω as in the preceding sections. We consider R ∈ Top (N) b and we assume T WSBC (R) ∅, as in Lemma 14.5. We let R ′ ∈ T WSBC (R) be such that ℓ(R ′ ) = 2 −k 0 ℓ(R), with k 0 = k 0 (γ) ≥ 1 big enough. Given λ > 0 and i = 1, 2, we set Here we are assuming that the corkscrews z i Q belong to the set V i from Lemma 14.5 and that λ is small enough.
Lemma 15.2 (Baby Key Lemma). Let p, R 0 , R, R ′ be as above. Given λ > 0, define also H i (R ′ ) as above. For a given τ > 0, suppose that If γ is small enough in the definition of V i in Lemma 14.5 (depending on τ and λ), then Remark that Γ depends on γ (see Lemma 14.5), and thus the families WSBC(Γ), Stop WSBC (R), H i (R ′ ) also depend on γ. The reader should thing that Γ → ∞ as γ → 0.
A key fact in this lemma is that the constants λ, τ can be taken arbitrarily small, without requiring ε → 0 as λτ → 0. Instead, the lemma requires γ → 0, which does not affect the packing condition in Lemma 13.2.
We denote with W P as in the Lemma 14.5. That is, W P the family of Whitney cubes I ⊂ V 1 ∪V 2 such that 1.1I ∩ ∂(V 1 ∪ V 2 ) ∩ 2B P ∅. So the family Bdy(R ′ ) contains Whitney cubes which intersect the boundaries of V 1 or V 2 and are close to 10B R ′ . Let us introduce some extra piece of notation. Given q ∈ R n+1 and 0 < r < s we let A(q, r, s) = B(q, s) \ B(q, r).
To prove Lemma 15.2, first we need the following auxiliary result.
Lemma 15.3. Let p, R 0 , R, R ′ be as above and, for i = 1 or 2, let Q ∈ H i (R ′ ). Let V i be as in Lemma 14.5 and let q ∈ Ω be a corkscrew point for Q which belongs to V i . Denote r = 2ℓ(R ′ ) and for δ ∈ (0, 1/100) set Let us note that the fact that q is a corkscrew for Q contained in V i implies that dist(q, ∂V i ) ≈ ℓ(Q), by the construction of the sets V i in Lemma 14.5.
So ϕ is smooth, and it satisfies 2I.
Observe that, in a sense, ϕ is a smooth version of the function χ B(q,r)∩V 1 .
From the above estimates we infer that g(p, q) ≤ |I 1 + I 3 | + c ε g(p, q).
Since neither I 1 nor I 3 depend on ε, letting ε → 0 we get A r,δ = x ∈ A(q, 1.2, 1.8r) ∩ V 1 \ F : δ Ω (x) ≤ δ r . Next we split the last integral as follows: Concerning J 1 , we have Thus, using also that |∇ϕ| 1/r outside F, Regarding J 2 , using Cauchy-Schwarz, we get To estimate the integral A r,δ g(p, x) 2 dx, we take into account that, for all x ∈ A r,δ ,  A(q,r,2r) g(q, x) dx.
By interchanging, p and q, it is immediate to check that an analogous estimate holds for the second summand on the right hand side of (15.6). Thus we get (15.7) J 2 δ α/2 r n+3 A(q,r,2r) g(p, x) dx Since r ≈ ℓ(R ′ ), we derive Concerning the sum S 1 we have Next we take into account that , where x I stands for the center of I. Then we derive Since Q∈ H 1 (R ′ ) ω x I (4Q) 1 for each I, we get .
By Lemma 14.5(e)(ii), we have I∈W P ℓ(I) n ℓ(P) n , and so we deduce Next we turn our attention to the sum S 2 in (15.9). Recall that Let us remark that we assume the condition that I ∈ W P for some 2P ∈ End(R ′ ) such that 2B P ∩ 10B R ′ ∅ to be part of the definition of I ∈ T 2 . Using the estimate m 4I g(p, ·) ω p (B I ) ℓ(I) 1−n , we derive To estimate the term A we take into account that if 20I ∩20B Q ∅ and I ∈ W P , then ℓ(P) ℓ(Q) and thus ℓ(I) γ 1/2 ℓ(Q) because I ∈ T 2 . As a consequence, I ⊂ 21B Q and also, by the Hölder continuity of g(z 1 Q , ·), if we let B be a ball concentric with B I with radius comparable to ℓ(Q) and such that dist(z 1 Q , B) ≈ ℓ(Q), we obtain m 2B I g(z 1 Q , ·) r(B I ) r(B) α m B g(z 1 Q , ·) γ α/2 1 ℓ(Q) n−1 , where α > 0 is the exponent of Hölder continuity. Hence, A γ α/2 Q∈ H 1 (R ′ ) P∈ End(R ′ ): 2B P ∩10B R ′ ∅ 20B P ∩20B Q ∅ I∈W P ∩T 2 ω p (B I ).
By Lemma 14.5(e)(ii), we have I∈W P ω p (B I ) ω p (CB P ), and using also that, for P as above, CB P ⊂ C ′ B Q for some absolute constant C ′ , we obtain Finally, we turn our attention to the term B. We have We claim now that, in the last sum, if one assumes that 20I ∩ 20B Q = ∅, then dist(I, 8B Q ) ≥ c γ −1/2 ℓ(I). To check this, take P ∈ End(R ′ ) such that I ∈ W P . Then note that ℓ(P) ≤ 1 300 d R (x P ) ≤ 1 300 dist(x P , Q) + ℓ(Q) ≤ 1 300 dist(x P , I) + diam(I) + dist(I, 8B Q ) + Cℓ(Q) .
Proof. For any R ′ ∈ D k 0 ∩ T WSBC (R), with k 0 = k 0 (γ), we define H i (R ′ ) as in (15.1), so that Stop WSBC (R) ∩ G ∩ D(R ′ ) = H 1 (R ′ ) ∪ H 2 (R ′ ). For each R ′ , we set That is, for fixed i = 1 or 2, if P∈H i (R ′ ) σ(P) ≤ τ σ(R ′ ), then all the cubes from H i (R ′ ) belong to Ex WSBC (R ′ ). In this way, it is clear that We claim that the λ-good corkscrews of cubes from Stop WSBC (R) ∩ G ∩ D(R ′ ) \ Ex WSBC (R ′ ) can be joined to some λ-good corkscrew for R ′ by a C-good Harnack chain, with λ depending on λ, η, and C depending on Γ and thus on λ, η too. Indeed, if Q ∈ H i (R ′ ) \ Ex WSBC (R ′ ) and z i Q is λ-good corkscrew belonging to V i (we use the notation of Lemma 15.2 and 14.5), then P∈H i (R ′ ) σ(P) > τ σ(R ′ ) by the definition above and thus Lemma 15.2 ensures that g(p, z i R ′ ) ≥ c(λ, τ) ℓ(R ′ ) σ(R 0 ) . So z i R ′ is a λ-good corkscrew, which by Lemma 14.5(c) can be joined to z i Q by a C-good Harnack chain. In turn, this λ-good corkscrew for R ′ can be joined to some λ ′ -good corkscrew for R by a C ′ -good Harnack chain, by applying Lemma 13.6 k 0 times, with C ′ depending on k 0 and thus on λ and η.
On the other hand, the cubes Q ∈ Stop WSBC (R) ∩ G which are not contained in any cube R ′ ∈ D k 0 ∩ T WSBC (R) satisfy ℓ(Q) ≥ 2 −k 0 , and then, arguing as above, their associated λ-good corkscrews can be joined to some λ ′ -good corkscrew for R by a C ′ -good Harnack chain, by applying Lemma 13.6 at most k 0 times. Hence, if we define taking into account (15.11), the lemma follows.
Proof of the Key Lemma 13.5. We choose Γ = Γ(λ, η) as in Lemma 15.10 and we consider the associated family WSBC(Γ). In case that T WSBC (R) = ∅, we set Ex(R) = ∅. Otherwise, we consider the family Ex WSBC (R) from Lemma 15.10, and we define It may be useful for the reader to compare the definition above with the partition of Stop(R) in (13.7). By Lemma 15.10 we have P∈Ex(R) σ(P) ≤ Q∈Ex WSBC (R) σ(P) ≤ η σ(R).
Next we show that for every P ∈ Stop(R) ∩ G \ Ex(R), any λ-good corkscrew for P can be joined to some λ ′ -good corkscrew for R by a C(λ, η)-good Harnack chain. In fact, if P ∈ Stop WSBC (R), then P ∈ Stop WSBC (R) ∩ G \ Ex WSBC (R) since such cube P cannot belong to SubStop(Q) for any Q ∈ Stop WSBC (R) \ Stop(R) (recall the partition (13.7)), and thus the existence of such Harnack chain is ensured by Lemma 15.10. On the other hand, if P Stop WSBC (R), then P is contained in some cube Q(P) ∈ Stop WSBC (R) \ WSBC(Γ). Consider the chain P = S 1 ⊂ S 2 ⊂ · · · ⊂ S m = Q(P), so that each S i is the parent of S i−1 . For 1 ≤ i ≤ m, choose inductively a big corkscrew x i for S i in such a way that x 1 is at the same side of L P as the good λ corkscrew z P for P, and x i+1 is at the same side of L S i as x i for each i. Using that bβ(S i ) ≤ Cε ≪ 1 for all i, it easy to check that the line obtained by joining the segments [z P , x 1 ], [x 1 , x 2 ],. . . ,[x m−1 , x m ] is a good carrot curve and so gives rise to a good Harnack chain that joins z P to x m . It may happen that x m is not a λ-good corkscrew. However, since Q(P) WSBC(Γ), it turns out that x m can be joined to some c 3 -good corkscrew z Q(P) for Q(P) by some C(Γ)-good Harnack chain, with c 3 given by (12.5) (and thus independent of λ and η), because Q(P) ∈ G. Note that since λ ≤ c 3 , z Q(P) is also a λ-good corkscrew. In turn, since Q(P) Ex WSBC (R), z Q(P) can be joined to some λ ′ -good corkscrew for R by another C ′ (λ, η)-good Harnack chain. Altogether, this shows that z P can be connected to some λ ′ -good corkscrew for R by a C ′′ (λ, η)-good Harnack chain, which completes the proof of the lemma.
Below we will write Ex(R, λ, η) instead of Ex(R) to keep track of the dependence of this family on the parameters λ and η.
16. Proof of the Main Lemma 10.2 16.1. Notation. Recall that by the definition of G K 0 in (13.4), R∈Top χ R (x) ≤ K for all x ∈ G K 0 . For such x, let Q be the smallest cube from Top that contains x, and denote n 0 (x) = − log 2 ℓ(Q), so that Q ∈ D n 0 (x) . Next let N 0 ∈ Z be such that σ x ∈ G K 0 : n 0 (x) ≤ N 0 − 1 ≥ 1 2 σ(G K 0 ), and denote G K 0 = x ∈ G K 0 : n 0 (x) ≤ N 0 − 1 . were defined in Section 13.2). So if R ∈ T ′ a \ D N (R 0 ), then Stop N (R) coincides the family of sons of R, and it R ∈ T ′ b this will not be the case, in general. Next we denote by T a and T b the respective subfamilies of cubes from T ′ a and T ′ b which intersect G K 0 . For j ≥ 0, we set We also denote and we let T j a be the subfamily of cubes R ∈ T a such that there exists some Q ∈ S j−1 b such that Q ⊃ R and R is not contained in any cube from S k b with k ≥ j.
Lemma 16.1. The following properties hold for the family T 1 b : (a) The cubes from T 1 b are pairwise disjoint and cover G K 0 , assuming N 0 big enough. (b) If R ∈ T 1 b , then ℓ(R) ≈ K ℓ(R 0 ). (c) Given R ∈ D(R 0 ) with ℓ(R) ≥ c ℓ(R 0 ) (for example, R ∈ T 1 b ) and λ > 0, if z R is a λ-good corkscrew point for R, then there is a C(λ, c)-good Harnack chain that joins z R to p.
Proof. Concerning the statement (a), the cubes from T 1 b are pairwise disjoint by construction. Suppose that x ∈ G K 0 is not contained in any cube from T 1 b . By the definition of the family Top N , this implies that all the cubes Q ⊂ R 0 with 2 −N ℓ(R 0 ) ≤ ℓ(Q) ≤ 2 −10 ℓ(R 0 ) containing x belong to T a . However, there are at most K cubes Q of this type, which is not possible if N is taken big enough. So the cubes from T 1 b cover G K 0 . and thus This finishes the proof of the Main Lemma 10.2.

Appendix A. Some counter-examples
We shall discuss some counter-examples which show that our background hypotheses in Theorem 1.3 (namely, n-ADR and interior corkscrew condition) are in some sense in the nature of best possible. In the first two examples, Ω is a domain satisfying an interior corkscrew condition, such that ∂Ω satisfies exactly one (but not both) of the upper or the lower n-ADR bounds, and for which harmonic measure ω fails to be weak-A ∞ with respect to surface measure σ on ∂Ω. In this setting, in which full n-ADR fails, there is no established notion of uniform rectifiability, but in each case, the domain will enjoy some substitute property which would imply uniform rectifiability of the boundary in the presence of full n-ADR.
In the last example, we construct an open set Ω with n-ADR boundary, and for which ω ∈ weak-A ∞ with respect to surface measure, but for which the interior corkscrew condition fails, and ∂Ω is not n-UR.
Failure of the upper n-ADR bound. In [AMT1], the authors construct an example of a Reifenberg flat domain Ω ⊂ R n+1 for which surface measure σ = H n ⌊ ∂Ω is locally finite on ∂Ω, but for which the upper n-ADR bound (A.1) σ(∆(x, r) ≤ Cr n fails, and for which harmonic measure ω is not absolutely continuous with respect to σ. Note that the hypothesis of Reifenberg flatness implies in particular that Ω and Ω ext := R n+1 \ Ω are both NTA domains, hence both enjoy the corkscrew condition, so by the relative isoperimetric inequality, the lower n-ADR bound (A.2) σ(∆(x, r) ≥ cr n holds. Thus, it is the failure of (A.1) which causes the failure of absolute continuity: in the presence of (A.1), the results of [DJ] apply, and one has that ω ∈ A ∞ (σ), and that ∂Ω satisfies a "big pieces of Lipschitz graphs" condition (see [DJ] for a precise statement), and hence is n-UR. We note that by a result of Badger [Bad], a version of the Lipschitz approximation result of [DJ] still holds for NTA domains with locally finite surface measure, even in the absence of the upper n-ADR condition.
Failure of the lower n-ADR bound. In [ABHM,Example 5.5], the authors give an example of a domain satisfying the interior corkscrew condition, whose boundary is rectifiable (indeed, it is contained in a countable union of hyperplanes), and satisfies the upper n-ADR condition (A.1), but not the lower n-ADR condition (A.2), but for which surface measure σ fails to be absolutely continuous with respect to harmonic measure, and in fact, for which the non-degeneracy condition fails to hold uniformly for x ∈ Ω, for any fixed positive η and c, and therefore ω cannot be weak-A ∞ with respect to σ. We note that in the presence of the full n-ADR condition, if ∂Ω were contained in a countable union of hyperplanes (as it is in the example), then in particular it would satisfy the "BAUP" condition of [DS2], and thus would be n-UR [DS2,Theorem I.2.18,p. 36].
Failure of the interior corkscrew condition. The example is based on the construction of Garnett's 4-corners Cantor set C ⊂ R 2 (see, e.g., [DS2,Chapter 1]). Let I 0 be a unit square positioned with lower left corner at the origin in the plane, and in general for each k = 0, 1, 2, . . . , we let I k be the unit square positioned with lower left corner at the point (2k, 0) on the x-axis. Set Ω 0 := I 0 .
Let Ω 1 be the first stage of the 4-corners construction, i.e., a union of four squares of side length 1/4, positioned in the corners of the unit square I 1 , and similarly, for each k, let Ω k be the k-th stage of the 4-corners construction, positioned inside I k . Note that dist(Ω k , Ω k+1 ) = 1 for every k. Set Ω := ∪ k Ω k . It is easy to check that ∂Ω is n-ADR, and that the nondegeneracy condition (A.3) holds in Ω for some uniform positive η and c, and thus by the criterion of [BL], ω ∈ weak-A ∞ (σ). On the other hand, the interior corkscrew condition clearly fails to hold in Ω (it holds only for decreasingly small scales as k increases), and certainly ∂Ω cannot be n-UR: indeed, if it were, then ∂Ω k would be n-UR, with uniform constants, for each k, and this would imply that C itself was n-UR, whereas in fact, as is well known, it is totally non-rectifiable. One can produce a similar set in 3 dimensions by simply taking the cylinder Ω ′ = Ω × [0, 1]. Details are left to the interested reader.