Uniform rectifiability implies Varopoulos extensions

We construct extensions of Varopolous type for functions $f \in \text{BMO}(E)$, for any uniformly rectifiable set $E$ of codimension one. More precisely, let $\Omega \subset \mathbb{R}^{n+1}$ be an open set satisfying the corkscrew condition, with an $n$-dimensional uniformly rectifiable boundary $\partial \Omega$, and let $\sigma := \mathcal{H}^n\lfloor_{\partial \Omega}$ denote the surface measure on $\partial \Omega$. We show that if $f \in \text{BMO}(\partial \Omega,d\sigma)$ with compact support on $\partial \Omega$, then there exists a smooth function $V$ in $\Omega$ such that $|\nabla V(Y)| \, dY$ is a Carleson measure with Carleson norm controlled by the BMO norm of $f$, and such that $V$ converges in some non-tangential sense to $f$ almost everywhere with respect to $\sigma$. Our results should be compared to recent geometric characterizations of $L^p$-solvability and of BMO-solvability of the Dirichlet problem, by Azzam, the first author, Martell, Mourgoglou and Tolsa and by the first author and Le, respectively. In combination, this latter pair of results shows that one can construct, for all $f \in C_c(\partial \Omega)$, a harmonic extension $u$, with $|\nabla u(Y)|^2 \text{dist}(Y,\partial \Omega) \, dY $ a Carleson measure controlled by the BMO norm of $f$, only in the presence of an appropriate quantitative connectivity condition.


List of symbols C µ
Carleson norm of the measure µ (Definition 2.4) C A Carleson packing norm of A ⊂ D (Definition 2.23) D collection of dyadic cubes (Theorem 2.16) Γ(x) dyadic cone at x ∈ ∂Ω (Definition 7.3) Γ(x) cone at x ∈ ∂Ω (Definition 2.1) semi-closed truncated cone at x ∈ Q ⊂ ∂Ω (Section 4) open set in R n+1 with ADR boundary ∂Ω ω X harmonic measure with pole at X ∈ Ω U Q , U Q dilated and non-dilated closed Whitney region (Sections 4 and 7)

Introduction
Connections between boundary geometry and PDE estimates have been studied for a long time (see e.g. the seminal work of F. and M. Riesz [RR20]) but the work is still ongoing and active. In the last couple of years, a lot of progress has been made, particularly in domains with codimension 1 Ahlfors-David regular (ADR) or uniformly rectifiable (UR) boundaries (see [Hof19] for a survey of some of these recent advances). In this article, we complement recent results related to geometric characterizations of solvability of Dirichlet problems, by showing that an extension property for BMO functions, first proved by Varopoulos in the half-space [Var77,Var78], remains true even in settings where harmonic extension of BMO boundary data (i.e., BMO-solvability of the Dirichlet problem) may fail: in fact, we show in the present paper that the Varopoulos extension property holds always for UR sets of codimension 1. In particular, our results do not require any kind of connectivity hypothesis on the domain or its boundary, whereas the analogous PDE solvability results cannot hold without certain quantitative connectivity assumptions.
Let us be more precise. Recently, Azzam, the first author, Martell, Mourgoglou and Tolsa [AHM + 19] have presented a geometric characterization of quantitative scale-invariant absolute continuity (i.e. the weak-A ∞ property) of harmonic measure with respect to the surface measure. Their result together with recent work of the first author and Le [HL18] gives us the following characterization theorem. For definitions of the properties mentioned in the theorem and in the rest of the introduction, see Section 2.
Theorem ([AHM + 19, HL18]). Let Ω ⊂ R n+1 be an open set satisfying the corkscrew condition and suppose that ∂Ω is n-ADR. Then the following conditions are equivalent: (1) ∂Ω is UR and Ω satisfies the weak local John condition, (2) harmonic measure belongs to the class weak-A ∞ with respect to the surface measure σ := H n ∂Ω on ∂Ω, (3) the Dirichlet problem is L p -solvable for some p < ∞, (4) the Dirichlet problem is BMO-solvable.
By L p -solvability we mean that there exists a constant C such that if f ∈ L p (∂Ω), then the solution u to the Dirichlet problem with data f converges non-tangentially to f and where N * is a non-tangential maximal operator. Many key results related to this concept can be found in the monograph of Kenig [Ken94]. By BMO solvability 1 , we mean that there exists a constant C such that if f is a compactly supported continuous function on ∂Ω, then the solution u to the Dirichlet problem satisfies the Carleson measure estimate sup x∈∂Ω,0<r diam(∂Ω) where ∆(x, r) := B(x, r) ∩ ∂Ω. This type of solvability was first shown to be equivalent to L p -solvability, for some p < ∞, by Dindos, Kenig and Pipher [DKP11], in Lipschitz or chord-arc domains (see also [Zha18] for an extension to 1-sided chord-arc domains). It was previously known that the weak-A ∞ property of harmonic measure (equivalently, L p -solvability for some p < ∞) may fail in the absence of connectivity, even if the boundary is UR [BJ90], but the result of [AHM + 19] is the first that tells us precisely how much connectivity we need (although we refer the reader to related work of Azzam [Azz18], concerning the analogous geometric characterization problem, in the case that harmonic measure is doubling). In particular, there are many domains with ADR or even UR boundaries for which one does not have BMO-solvability, nor L p -solvability for any finite p.
In this work, we nonetheless obtain extension results of Varopoulos type that can be seen as substitutes for these solvability theorems, in domains with n-UR boundaries, but in which the weak local John property may fail. We first consider extensions of L ∞ functions: Theorem 1.1. Let Ω ⊂ R n+1 be an open set satisfying the corkscrew condition, with n-UR boundary. Then for every Borel measurable f ∈ L ∞ (∂Ω, dσ), there is a function Φ = Φ f in Ω, such that i) Φ ∈ C ∞ (Ω), and |∇Φ(X)| ≤ C f L ∞ (∂Ω) δ(X) −1 , for all X ∈ Ω.
for σ-a.e. x ∈ ∂Ω, iv) |∇Φ(Y )| dY is a Carleson measure: The definition is slightly different if Ω is unbounded and ∂Ω is bounded; see [HL18, Section 5] for details. Proposition 1.3. Suppose that Ω ⊂ R n+1 is an open set satisfying the corkscrew condition with d-ADR boundary for some d ∈ (0, n]. Let D be a dyadic system on ∂Ω, Q 0 ∈ D be a fixed dyadic cube and {Q j } j ⊂ D Q 0 be a collection of subcubes of Q 0 . Suppose that function f in ∂Ω, f (x) = j α j 1 Q j , satisfies the following conditions: • f ∈ BMO(∂Ω), • there exists C 0 ≥ 1 such that Q j ⊂Q σ(Q j ) ≤ C 0 σ(Q) for every Q ∈ D, • sup j |α j | f BMO .
We remark that in proving Theorem 1.2, we shall use only the codimension 1 case (i.e., d = n) of Proposition 1.3.
Unlike that of Theorem 1.1, the proof of Proposition 1.3 does not require any UR machinery. Many of the key arguments are fairly elementary but still a bit delicate. A principal difficulty is the need to build suitable substitutes for Carleson boxes that are compatible with non-tangential convergence, as well as with proving the Carleson measure estimate (1.4). Both the construction of our boxes and the rest of our techniques work for d-ADR boundaries for any d ∈ (0, n], including non-integer dimensions. We conjecture that if Ω ⊂ R n+1 is an open set satisfying the corkscrew condition with n-ADR boundary ∂Ω, then the existence of extensions (with some suitable convergence to the boundary values) as in Theorem 1.1 implies that ∂Ω is n-UR. We note that if these extensions exist, then also extensions as in Theorem 1.2 exist since Proposition 1.3 and Lemma 10.1 hold with just the ADR assumption.
The paper is organized as follows. In the next section, we discuss the basic notation and definitions in the paper. In Section 3, we consider ε-approximators and many regularization lemmas we need later. We build machinery for Theorem 1.1 in Sections 4 and 5, and we prove the theorem in Section 6. In Sections 7 and 8, we revisit and modify the construction of Whitney regions and Carleson boxes and we use the modified construction to prove Proposition 1.3 in Section 9. Finally, in Section 10, we prove a version of Garnett's decomposition lemma and combine it with Theorem 1.1 and Proposition 1.3 to prove Theorem 1.2.

Notation and basic definitions
We use the following notation.
• Ω ⊂ R n+1 will always be an open set with non-empty d-dimensional ADR boundary ∂Ω (see Definition 2.11). In Sections 4, 5, and 6, we additionally assume that ∂Ω is n-UR (see Definition 2.13) and that Ω satisfies the corkscrew condition (see Definition 2.12). • The letters c and C denote constants that depend only on dimension, ADR constant (see Definition 2.11), UR constants (see Definition 2.13) and other similar parameters. The values of c and C may change from one occurence to another. We do not track how our bounds depend on these constants and usually just write γ 1 γ 2 if γ 1 ≤ cγ 2 for a constant like this c and γ 1 ≈ γ 2 if γ 1 γ 2 γ 1 . If the constant c κ depends only on parameters of the previous type and some other parameter κ, we usually write γ 1 κ γ 2 instead of γ 1 ≤ c κ γ 2 .
• We use capital letters X, Y, Z, and so on to denote points in Ω and lowecase letters x, y, z, and so on to denote points in ∂Ω. • The (n + 1)-dimensional Euclidean open ball of radius r will be denoted B(x, r) or B(X, r) depending on whether the center point lies on ∂Ω or Ω. We denote the surface ball of radius r centered at x by ∆(x, r) := B(x, r) ∩ ∂Ω. • Given a Euclidean ball B := B(X, r) or a surface ball ∆ := ∆(x, r) and constant κ > 0, we denote κB := B(X, κr) and κ∆ := ∆(x, κr). • For every X ∈ Ω we set δ(X) := dist(X, ∂Ω).
• We let H d be the d-dimensional Hausdorff measure and denote the surface measure of ∂Ω by σ := H d ∂Ω . The (n + 1)-dimensional Lebesgue measure of a measurable set A ⊂ Ω will be denoted by |A|.
• For a set A ⊂ R n+1 , we let 1 A be the indicator function of A: • The interior of a set A will be denoted int(A).
• The unit outer normal (when it exists) will be denoted by − → N . • For µ-measurable sets A with positive and finite measure we set f A := ffl A f dµ := 1 µ(A) f dµ. Definition 2.1 (Cones and non-tangential limits). Suppose that m > 1. For every x ∈ ∂Ω, the cone of m-aperture at x is the set Let G be a function defined in Ω, g be a function defined on ∂Ω and x be a point on ∂Ω. We consider two types of non-tangential convergence in this paper. We use the notation lim Y →x N.T. G(Y ) = g(x) for both of them, but the meaning should be clear from context.
• With standard type non-tangential convergence we mean that there exists m > 1 such that we have • With one-sided non-tangential convergence we mean that there exists m > 1 and a connected component is an open set satisfying the corkscrew condition, with UR boundary ∂Ω, then for σ-a.e. x ∈ ∂Ω, the cone with vertex at x has at most two connected components inside Ω such that their boundaries contain x, by Lemma 4.13 (see also Lemma 4.14, and Remarks 4.15 and 4.16). ii) In the actual calculations related to non-tangential convergence, we use dyadic cones that we define in later sections (see Section 4 and Section 7). These dyadic cones always contain a truncated cone of the type Γ(x), at least locally.
Definition 2.3 (BMO and dyadic BMO). The space BMO(∂Ω) (bounded mean oscillation) consists of those locally integrable function f such that where the supremum is taken over all surface balls ∆ ⊂ ∂Ω. We define the dyadic BMO space BMO D (∂Ω) by replacing the supremum over all surface balls with the supremum over all dyadic cubes Q (see Theorem 2.16).
Definition 2.4 (Carleson measures). We say that a Borel measure µ in Ω is a Carleson measure (with respect to ∂Ω) if we have We call C µ the Carleson norm of µ.
Definition 2.6 (Local BV). We say that locally integrable function f has locally bounded variation in Ω (denote f ∈ BV loc (Ω)) if for any open relatively compact set Ω ⊂ Ω the total variation over Ω is finite: where C 1 0 (Ω ) is the class of compactly supported continuously differentiable vector fields in Ω .
Definition 2.8 (Weak local John condition). We say that Ω satisfies the weak local John condition if there exist constants λ ∈ (0, 1), θ ∈ (0, 1] and R ≥ 2 such that for every X there exists a Borel set F ⊂ ∆ X := B(X, Rδ(X)) ∩ ∂Ω such that σ(F ) ≥ θσ(∆ X ) and for every y ∈ F there is a λ-carrot path connecting y to X. Definition 2.9 (Weak A ∞ ). Let ν be a measure defined on ∂Ω and ∆ 0 := B 0 ∩ ∂Ω be a surface ball. We say that ν belongs to weak-A ∞ (∆ 0 ) if there are positive constants C and s such that for each surface ball ∆ := B ∩ ∂Ω centered on ∂Ω with B ⊂ B 0 we have for every Borel set A ⊂ ∆.
We note that the constant 2 in (2.10) can be replaced with any constant c > 1 without changing the class weak-A ∞ (∆ 0 ) (see e.g. [AHT17, Section 8]).
Definition 2.11 (ADR). We say that a closed set E ⊂ R n+1 is a d-ADR (Ahlfors-David regular) set for d ∈ (0, n] if there exists a uniform constant C such that for every x ∈ E and every r ∈ (0, diam(E)), where diam(E) may be infinite.
As it is well-known, UR is a necessary and sufficient condition for many types of PDE and Calderón-Zygmund type harmonic analysis results on ADR sets or open sets with ADR boundaries. In this paper, we work with two characterizations UR: ε-approximability of harmonic function (see Section 3) and bilateral corona decomposition (see Section 4). We use ε-approximability to build the extension in Theorem 1.1, and the bilateral corona decomposition, and its consequences, as a tool to prove some convergence properties.
Definition 2.14 (NTA). Following [JK82], we say that a domain Θ ⊂ R n+1 is NTA (nontangentially accessible) if • Θ satisfies the Harnack chain condition: there exists a uniform constant C such that for every ρ > 0, Λ ≥ 1 and X, X ∈ Θ with δ(X), δ(X ) ≥ ρ and |X − X | < Λρ there exists a chain of open balls B 1 , . . . , • both Θ and R n+1 \ Θ satisfy the corkscrew condition. Theorem 2.16 (E.g. [Chr90,SW92,HK12]). Suppose that E is a d-ADR set. Then there exists a countable collection D (that we call a dyadic system), of Borel sets Q k α (that we call dyadic cubes) such that (i) the collection D is nested: if Q, P ∈ D, then Q ∩ P ∈ {∅, Q, P }, (ii) E = Q∈D k Q for every k ∈ Z and the union is disjoint, (iii) there exist constants c 1 > 0 and C 1 ≥ 1 such that for all cubes Q k α and for all ∈ (0, c 1 ). In addition, there exists a collection of dyadic systems {D ν } N ν=1 on E, of bounded cardinality N , and a uniform constant C, such that if ∆ = B ∩ E is any surface ball centered on E, then there is at least one choice of dyadic system D ν , and a cube Q ∈ D ν , with ∆ ⊂ Q, and with diam(Q) ≤ min(Cdiam(B), diam(E)).
Remark 2.19. In general spaces of homogeneous type, dyadic systems were first constructed in [Chr90] for some parameter δ ∈ (0, 1) instead of the dyadic parameter 1/2 (we may always choose δ = 1/2 by [HMMM14]). In the same context, the adjacent systems {D ν } N ν=1 were contructed in [HK12] (see also [HT14,Tap16] for an alternative construction and some additional approximation properties in geometrically doubling metric spaces). For the history of adjacent systems in R n , see [CU17, Section 3].
Notation 2.20. We shall use the following notational conventions.
(1) Since the boundary ∂Ω may be bounded or disconnected, we may encounter a situation where Q k α = Q l β although k = l. Thus, when we consider cubes Q k α ∈ D, we assume that C 1 2 −k ≤ diam(∂Ω) and the number k is maximal in the sense that there does not exist a cube Q l β ∈ D such that Q l β = Q k α for some l > k. Notice that the number k is bounded for each cube since the ADR condition excludes the presence of isolated points in ∂Ω.
(2) For each k, and for every cube Q k α := Q ∈ D k , we denote (Q) := 2 −k and x Q := z k α . We call (Q) the side length of Q, and x Q the center of Q.
(3) For every Q ∈ D, we denote the collection of dyadic subcubes of Q by D Q .
Remark 2.21. We record the following further observations.
(1) The following exterior variant of (2.18) in Theorem 2.16 also holds for every Q ∈ D: as may be seen by covering the exterior shell Ext (Q) by dyadic cubes of uniform side length ≈ (Q), each of which is a subcube of one of a uniformly bounded number of neighbors of Q with side length equal to that of Q. Applying (2.18) in each of these neighbors, we obtain (2.22). (2) By the ADR property and (2.17), we have σ(Q) ≈ (Q) d with implicit constants independent of Q, and σ( Q) σ(Q) for the dyadic parent of Q, that is, the cube Q containing Q, and belonging to the generation immediately preceeding that of Q, (i.e., Q ∈ D k−1 when Q ∈ D k ). Similarly we have σ(κQ) κ σ(Q) for all κ > 1.
Definition 2.23. We say that a collection A ⊂ D satisfies a Carleson packing condition if there exists a constant C ≥ 1 such that for every cube Q 0 ∈ D. We call the smallest such constant C the Carleson packing norm of A and denote it by C A .
Lemma 2.24. Suppose E ⊂ R n+1 is a d-ADR set and that A ⊂ D satisfies a Carleson packing condition. Then we have for every cube Q ∈ D and every d ≤ n.
Proof. For d = n, in the presence of the n-ADR condition, the lemma is a trivial reformulation of Definition 2.23. Therefore let us suppose that d < n. In this case, the same trivial argument using d-ADR gives Q∈A,

ε-approximability and regularization
In the proof of Theorem 1.1, we follow the original idea of Varopoulos and construct the extension using ε-approximability of harmonic functions. It was recently shown that this property characterizes uniform rectifiability: in Ω is ε-approximable for every ε ∈ (0, 1): there exists a constant C ε and a function Φ = Φ ε ∈ BV loc (Ω) such that The direction UR implies ε-approximability appears in [HMM16], and the converse is proved in [GMT18]. (see also [HT17] and [BT19] for pointwise and L p versions of this result). For other characterizations of UR with respect to properties of harmonic functions or solutions to other elliptic PDE, see [HMM16, HMM19, GMT18, HT17, BT19, AGMT16].
Since ε-approximators are a crucial ingredient in the proof of Theorem 1.1, it will be convenient for us to use regularized ε-approximators that are locally Lipschitz: . We shall verify this lemma by a fairly straightforward mollifier argument (see e.g. [EG92, Section 4])). Since we need to regularize also other functions in subsequent sections, we formulate the following lemmas in a fairly general way.
We start by noting that although our distance function δ is Lipschitz, that is usually the best level of regularity we can hope for in this context. However, we can use a classical result of Stein to replace δ with a smooth function that is pointwise close to δ: . Let E ⊂ R n+1 be a closed set and δ E be the distance function with respect to E. Then there exist positive constants m 1 and m 2 and a function β E defined in E c such that In addition, the constants m 1 , m 2 and C α are independent of E.
Set Ω := R n+1 \ E, so that ∂Ω = E, and observe that for given X ∈ Ω, by construction. Suppose that G 0 : Ω → R is a locally integrable function. We set We then have the following.
for every X ∈ Ω, where C µ is the constant in (2.5).
Proof. We begin with some preliminary observations. With B X defined as in (3.4), note that by Theorem 3.3 and construction, (3.10) Moreover,˜Λ(X, Y ) dY = 1, for every X ∈ Ω, and therefore where we have used also (3.4), and Poincaré's inequality for BV (see [EG92, Theorem 1, p. 189]). Now letx ∈ ∂Ω be a "touching point" for X, i.e. |X −x| = δ(X). Then by hypothesis. Combining the latter estimate with (3.12), we obtain the desired conclusion. We remark that the full strength of the Carleson measure condition was not required here, but only the weaker estimate Proof. Fix B(z, r) with z ∈ ∂Ω. We cover B(z, r)∩Ω by (possibly disconnected) "half-open" regions Thus, using (3.12), we see thaẗ Summing in k, and using that the sets V * k have bounded overlaps, we obtain as desired.
Lemma 3.14. If G 0 converges to g(x) non-tangentially in the standard sense (respectively, in the one-sided sense) in a cone with large enough aperture, then also G converges to g(x) non-tangentially in the standard (respectively, one-sided) sense.
Proof. Suppose that Y ∈ Γ m (x) for some m > 1. We recall that 2m 2 ), we can use the facts that˜ζ = 1 and ζ(X) ≤ 1 for every X ∈ Ω to show that By combining these two observations we see that if G 0 converges to g(x) non-tangentially in a cone with aperture m, then G converges to g(x) non-tangentially in a cone with aperture m − 1 2 . Observe that the preceding argument applies in the case of either standard or one-sided non-tangential convergence.
Remark 3.15. The aperture of the cones does not play an important role in this paper and we use Lemma 3.14 without considering details related to them in the proofs. This is because we can always use mollifiers that are supported on a smaller ball than B(0, 1 2m 2 ) and we use dyadic cones that we can construct in such a way that they contain cones of the type Γ m for a large m (see Section 7).

Bilateral corona decomposition and one-sided non-tangential traces
In R n+1 + , the construction of dyadic Carleson boxes and dyadic Whitney regions is very simple: just take a dyadic cube on R n , build a cube on top of it to get the Carleson box and remove the lower half of the cube to get the Whitney region. These objects are easy to work with particularly due to their simple geometric structure and they are very effective in many situations (see e.g. [HKMP15,HR18]). However, it is still possible to construct substitutes for these boxes and regions that share many good properties with their R n+1 + -analogues [HMM16, Section 3]. In this paper, we need two versions of the Whitney regions from [HMM16] for two different purposes: 1) the original regions in a slightly modified form to prove Theorem 1.1 in Section 6, 2) simplified and non-dilated regions for the construction of the extension of Proposition 1.3.
The reason why we need these simplified regions is that although the boundaries of the original dilated regions are ADR, they are not quite neat enough for some more delicate estimates. We construct these regions in Sections 7 and 8.
Let us start by recalling some key tools from [HMM16]. In this section, Ω ⊂ R n+1 is an open set with n-UR boundary ∂Ω and D is a dyadic system on ∂Ω. We begin with a standard Whitney decomposition of Ω.
4.1. Whitney cubes and regions. We use Whitney cubes and Whitney regions in our proofs and constructions throughout the article. Suppose that W := {I} I is a Whitney decomposition of Ω (see e.g. [Ste70,Chapter VI], that is, {I} I is a collection of closed (n + 1)-dimensional Euclidean cubes whose interiors are disjoint such that I I = Ω and whenever I 1 ∩ I 2 = ∅. For parameters η and K satisfying η 1 K and for every Q ∈ D(∂Ω) we set We note that W 0 Q is non-empty, for η chosen small enough, and K large enough, provided that Ω satisfies the corkscrew condition (see [HMM16,Section 3]). In particular, the latter is true when In the sequel, we shall always assume that η and K have been so chosen.
Definition 4.3. For ξ > 1 and every I ∈ W, we let I * be the concentric dilation of I: We note that if ξ is close enough to 1, (and we shall always choose it so), the fattened cubes I * have bounded overlaps, and retain the property that diam(I * ) ≈ dist(I * , ∂Ω). We shall refer to such values of ξ as allowable.
If we choose (as above) the parameters η, K and ξ in a suitable way, the collections I∈W Q I and I∈W Q I * , and certain variants of these collections, have strong geometric properties that we will formulate in the next lemmas and use in the subsequent sections.
Definition 4.4. We say that a subcollection S ⊂ D is coherent if the following three conditions hold.
(a) There exists a maximal element Q(S) ∈ S such that Q ⊂ S for every Q ∈ S. (b) If Q ∈ S and P ∈ D is a cube such that Q ⊂ P ⊂ Q(S), then also P ∈ S. (c) If Q ∈ S, then either all children of Q belong to S or none of them do. If S satisfies only conditions (a) and (b), then we say that S is semicoherent. (1) The "good" collection G is a disjoint union of coherent stopping time regimes S.
(2) The "bad" collection B and the maximal cubes Q(S) satisfy a Carleson packing condition: for every Q ∈ D we have (3) For every S, there exists an n-dimensional Lipschitz graph Γ S , with Lipschitz constant at most η, such that for every Q ∈ S we have Next, we recall a construction in [HMM16, Section 3], leading up to and including in particular [HMM16, Lemma 3.24]. We summarize this construction as follows.
Lemma 4.6. Let E ⊂ R n+1 be UR, and set Ω E := R n+1 \E. Given positive constants η 1 and K 1, as in (4.1) and Remark 4.2, let D = G ∪ B, be the corresponding bilateral Corona decomposition of Lemma 4.5. Then for each S ⊂ G, and for each Q ∈ S, the collection W 0 Q in (4.1) has an augmentation W * Q ⊂ W satisfying the following properties.
where (after a suitable rotation of coordinates) each I ∈ W * ,+ Q lies above the Lipschitz graph Γ S of Lemma 4.5, each I ∈ W * ,− Q lies below Γ S . Moreover, if Q is a child of Q, also belonging to S, then each I ∈ W * ,+ Q (resp. I ∈ W * ,− Q ) belongs to the same connected component of Ω E as each I ∈ W * ,+ Q (resp. I ∈ W * ,− Q ) and W * , (2) There are uniform constants c and C such that (4.7) (3) For ξ > 1, and recalling Definition 4.3, set and given S , a non-empty semi-coherent subregime of S, define Then there exists ξ 0 > 1 such that each of Ω ± S is a CAD (Definition 2.15), with chord-arc constants depending only on n, ξ, η, K, and the ADR/UR constants for E, provided that 1 < ξ < ξ 0 .
As in [HMM16], 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 4.5. Let W * Q be the augmentation of W 0 Q as constructed in Lemma 4.6, and set (4.10) For Q ∈ G we shall henceforth simply write W Q , W ± Q in place of W * Q , W * ,± Q . For arbitrary Q ∈ D, good or bad, we may then make the following definitions.
The closed Whitney region relative to Q, and its fattened version are, respectively, the sets Similarly, we define standard and fattened versions of the "semi-closed" (i.e., closed away from ∂Ω) truncated dyadic cone at x: and the "semi-closed" Carleson box relative to Q: We list some further properties of U Q and T Q in the next lemma. Most properties in the first lemma follow directly from the construction but some of them require slightly trickier estimates related to the choice of η and K and the bilateral corona decomposition (see [HMM16,Section 3

]).
For an open set Ω ⊂ R n+1 that satisfies an interior corkscrew condition and has ndimensional UR boundary ∂Ω, we define the Whitney regions U Q as above, but only include only those connected components contained in Ω (by the corkscrew condition, there must be at least one such). Of course, this includes the case that Ω = Ω E = R n+1 \ E, with for an n-dimensional UR set E = ∂Ω E , as in Lemma 4.6.
Lemma 4.12. Let Ω ⊂ R n+1 satisfy an interior corkscrew condition, with n-dimensional UR boundary ∂Ω. We have the following properties: • The region U Q is a union of a uniformly bounded number of Whitney cubes I such that (Q) ≈ (I) and dist(Q, I) ≈ (Q).
• If Q ∈ G, then U Q has at least one connected component, and at most two, corresponding to U ± Q in Lemma 4.6. • If Q ∈ B, then U Q has a uniformly bounded number of connected components.

4.2.
Non-tangential convergence of ε-approximators. We shall use the properties in Lemma 4.12 to prove some results about non-tangential convergence of ε-approximators.
Lemma 4.13. Let Ω ⊂ R n+1 be as in Lemma 4.12, and write D = B ∪ G as in Lemmas 4.5 and 4.6. Let Q 0 ∈ D be a fixed cube, denote for every x ∈ ∂Ω (thus G Q 0 vanishes outside of Q 0 ). Then G Q 0 (x) < ∞ for almost every x ∈ Q 0 . In particular, for almost every x ∈ Q 0 , there exists a stopping time regime S x such that if x ∈ Q and (Q) ≤ (Q(S x )), then Q ∈ S x . For each Q ∈ S x , the interior of the cone Υ Q (x) splits into at most two chord-arc domains, as does the sawtooth region Ω Sx .
Proof. Since the collection M Q 0 satisfies a Carleson packing condition by Lemma 4.5 and In particular, G Q 0 (x) < ∞ for almost every x ∈ Q 0 . Thus, for almost every x ∈ Q 0 there exist C x > 0 such that if x ∈ Q and (Q) < C x , then Q / ∈ M Q 0 . In particular, there exists a stopping time regime S x given by Lemma 4.5 such that if x ∈ Q and (Q) < C x , then Q ∈ S x ⊂ G. Thus, by Lemma 4.12, the corresponding Whitney region U Q splits into at most two connected components. The final property follows now from Lemma 4.6.
For every x ∈ ∂Ω that satisfies the condition in Lemma 4.13, we denote the components of Υ Q (x) by Υ ± Q (x), whose interiors, denoted by Υ ± Q (x), are subdomains of Ω ± Sx (see (4.9)), respectively. Since Ω satisfies the corkscrew condition, at least one of Ω ± Sx is contained in Ω, and it may be that both are. We define Υ +,fat Q(Sx) in the same way. Lemma 4.14.
Let Ω ⊂ R n+1 be an open set satisfying an interior corkscrew condition and let ∂Ω be UR. Suppose that Φ : Ω → R is a smooth function such that µ = |∇Φ(Y )| dY is a Carleson measure, and |∇Φ(X)| 1 δ(X) for every X ∈ Ω. Then Φ has one-sided non-tangential boundary traces in the following sense: for σ-a.e. x ∈ ∂Ω, the limits exist and satisfy ϕ ± L ∞ (∂Ω) ≤ Φ L ∞ (Ω) , provided that Ω ± Sx ⊂ Ω. Remark 4.15. As noted above, necessarily Ω ± Sx ⊂ Ω for at least one choice of + or −, and possibly both. Thus, Φ has at least a 1-sided non-tangential trace a.e. on ∂Ω. In the case that both components of Ω Sx are contained in Ω, the traces ϕ + and ϕ − may not coincide.
Proof of Lemma 4.14. Fix a cube Q 0 ∈ D, and let x ∈ ∂Ω be a point satisfying the condition G Q 0 (x) < ∞ in Lemma 4.13. We suppose that Ω + Sx ⊂ Ω, and consider the limit in Υ + Q(Sx) (x); the case that Ω − Sx ⊂ Ω may be handled by the same argument. Let {X k } k be an arbitrary sequence of points in Υ + Q(Sx) (x) such that X k → x. It suffices to show that {Φ(X k )} is a Cauchy sequence.
We have fixed 1 < ξ < ξ , and have constructed the corresponding standard and "fat" versions of the Whitney regions, cones and Carleson boxes as in Definition 4.11. Using Lemma 4.6, we set where C i is a cylinder with height h i and radius ρ i satisfying and such that dist(C i , ∂Ω) ≈ diam(C i ) ≈ r i , (vii) the balls {B i } i and the cylinders {C i } i have bounded overlaps. Here, the implicit constants depend on the NTA properties of Υ 0 , and possibly on ε.
We now have By the mean value theorem and the pointwise gradient bound, we know that Φ is locally Lipschitz. Thus, and similarly I 3 ε. As for I 2 , by (v) and (vi) above, and Poincaré's inequality, we have Then, by the bounded overlap property of the cylinders {C i } i ,and the structure of the dyadic cones, we have We notice that for σ-a.e. x ∈ ∂Ω. In particular, for σ-a.e. x ∈ ∂Ω. It follows that I 2 ≤ ε if k, m are large enough, and consequently that I 1 + I 2 + I 3 ε. We therefore conclude that {Φ(X k )} k is a Cauchy sequence.
Remark 4.16. As noted above (see Remark 4.15), it is possible that non-tangential traces, whose existence is guaranteed by Lemma 4.14, may exist from two sides, and they may not coincide. It will therefore be convenient to fix a canonical, unambiguous choice of non-tangential approach. To this end, we proceed as follows. Recall the counting function G Q defined in Lemma 4.13. Set Recall that for each cube Q, G Q (x) < ∞ for σ-a.e. x ∈ ∂Ω. Since D is countable, we find that σ(∂Ω \ A NT ) = 0. For each x ∈ A NT , there is a stopping time regime S x , as in Lemma 4.13, with maximal cube Q(S x ). We set D NT := {Q(S x )} x∈A NT , and observe that this collection is countable (thus, S x = S y for many choices of distinct x and y). We enumerate D NT = {Q i } ∞ i=1 , and for each Q i ∈ D NT , we let S i be the stopping time regime with maximal cube Q i . If Ω ± Sx is contained in Ω, then for every Φ as in Lemma 4.14, the non-tangential traces ϕ ± (x) are defined for σ-a.e. x ∈ A NT . In addition, for x ∈ A NT , there is an index i with S x = S i , and since the corkscrew condition holds in Ω, at least one of Ω ± S i is contained in Ω. If there is only one such, then the trace ϕ(x) is defined unambiguously; on the other hand, if both are contained in Ω, then we arbitrarily set ϕ(x) = ϕ + (x). Note that we make this same choice for every x ∈ A NT such that S x = S i , and moreover, that this choice is specified in advance, and is independent of Φ.

Some results on boundary behavior of bounded harmonic functions
In this section, we shall prove some useful facts about boundary behavior of bounded harmonic functions. We begin with some preliminary observations.
Remark 5.1. In the sequel, given a function v defined in an open set Ω, we let Tv denote the non-tangential trace of v on ∂Ω, i.e., for x ∈ ∂Ω, set provided that this non-tangential limit exists. Here, the notation Y → x N.T. means that Y → x, with Y ∈ Γ(x) (see Definition 2.1), or with Y ∈ Γ(x) (see Definition 7.3 below, and also Remark 7.5). We recall that in an NTA domain Ω, if v is a bounded harmonic function, then Tv(x) exists for ω-a.e. x ∈ ∂Ω, by virtue of the Fatou Theorem of [JK82, Theorem 6.4], where ω is harmonic measure for Ω with any fixed pole. Recall also that if, in addition, the NTA domain has an ADR boundary (i.e., so that Ω is a CAD; see Definition 2.15), then in particular, by results obtained independently in [DJ90] and in [Sem89], ω and σ = H n ∂Ω are mutually absolutely continuous, and thus for a bounded harmonic function v, one has that Tv(x) exists for σ-a.e. x ∈ ∂Ω. In particular, in this context, the Dirichlet problem is uniquely solvable in Ω, with data in L p (∂Ω, σ) for p < ∞ sufficiently large (depending on dimension and the chord-arc constants of Ω), with L p control of the non-tangential maximal function, and with non-tangential convergence of the solution to the data, σ-a.e. on ∂Ω. Therefore, in a bounded chord-arc domain Ω, if v is a bounded harmonic function with non-tangential trace Tv, we then have is also bounded in Ω. Then the non-tangential trace operator T satisfies the countable additivity property Proof. Set f := Tu , f k := Tu k , which, as noted above, exist σ-a.e. on ∂Ω, and of course inherit non-negativity from u and u k . Since T is a linear operator, for each positive integer N , and at σ-a.e. point on ∂Ω, where in the inequality we have used that u k ≥ 0 for every k. Letting N → ∞, we find that Our goal is then to show that f = f at σ-a.e. point on ∂Ω. To this end, since Ω is a bounded CAD, we may apply (5.3) to obtain for each Y ∈ Ω, where the interchange of summation and integration in the fourth equality may be justified by monotone convergence, since f k ≥ 0. Thus u = u at every point in Ω, hence, σ-a.e. on ∂Ω, we have In the sequel, given a set A, we denote the usual supremum norm of a function g defined on A by exists and is harmonic in Ω, and satisfies v sup(Ω) ≤ g sup(∂Ω) .
Our main result in this section is the following. Then the non-tangential trace Tv g exists σ-a.e. on ∂Ω, and Tv g (x) = g(x) , σ-a.e. x ∈ ∂Ω . (5.8) We note that no continuity assumption is imposed on g; moreover, in the generality of Lemma 5.6, harmonic measure need not be absolutely continuous with respect to surface measure on ∂Ω.
Proof. By Lemma 4.6 and Remark 4.16, there is a countable collection of bounded chordarc domains {Ω i } ∞ i=1 , with Ω i = Ω ± S i for some choice of ±, such that Ω i ⊂ Ω, and σ ∂Ω \ ∪ i ∂Ω i = 0 . (5.9) In the case that each of Ω ± S i is contained in Ω, then we may choose Ω i to be either of these. Moreover, by the Fatou theorem of [BH18], Tv g exists at σ-a.e. point on ∂Ω; more precisely, it exists at σ-a.e. point on ∂Ω ∩ ∂Ω i , for each i, as a one-sided non-tangential trace (i.e., with the limit taken through the non-tangential approach region within Ω i ); one may then invoke (5.9) to cover ∂Ω up to a set of σ-measure zero. Thus, it is enough to verify that (5.8) holds for σ-a.e. x ∈ ∂Ω ∩ ∂Ω 1 , where Ω 1 is any bounded chord-arc subdomain of Ω, whose boundary meets ∂Ω. We therefore fix such a subdomain Ω 1 , and let T 1 denote the non-tangential trace operator on ∂Ω 1 . Let g be an everywhere bounded Borel measurable function on ∂Ω, and define v g as in (5.7), so that v g is a bounded harmonic function in Ω.
We note that if x ∈ ∂Ω ∩ ∂Ω 1 is a point where Tv g (x) exists, then T 1 v g (x) exists, and since the non-tangential approach region in the subdomain Ω 1 is contained in a nontangential approach region for the ambient domain Ω. Observe also that since v g is, of course, continuous in Ω. Applying (5.3) in the bounded chord-arc domain where ω 1 is harmonic measure for Ω 1 . We also define v g (Y ) :=ˆ∂ Thus, by Remark 5.1, v g is the unique solution to the Dirichlet problem in Ω 1 with boundary data (T 1 v g )1 ∂Ω∩∂Ω 1 + v g 1 Ω∩∂Ω 1 , and v g is the unique solution to the Dirichlet problem in Ω 1 with boundary data g1 ∂Ω∩∂Ω 1 + v g 1 Ω∩∂Ω 1 . Moreover, each of these solutions converges non-tangentially in Ω 1 to its corresponding boundary data. In particular, σ 1 -a.e. on ∂Ω 1 , where σ 1 := H n ∂Ω 1 is the surface measure on ∂Ω 1 . We now claim that v g = v g in Ω 1 . Assuming the claim momentarily, we then have T 1 v g = T 1 v g , and this gives us T 1 v g (x) = g(x) for σ 1 -a.e. x ∈ ∂Ω 1 by (5.11). In particular, we have Tv g = g for σ-a.e. point on ∂Ω ∩ ∂Ω 1 by (5.10), and hence that (5.8) holds, as desired.
It therefore remains to verify that v g = v g in Ω 1 . To this end, we note first that the claim holds immediately in the special case that g is continuous on ∂Ω, since in that case Tv g = g at every point on ∂Ω (indeed, every boundary point is regular in the sense of Wiener, by the ADR property (see e.g. [HLMN17,Lemma 3.27] or [Zha18, Section 3])). By definition of v g and v g , we may write v g (Y ) =ˆ∂ For each Y ∈ Ω 1 , define two non-negative set functions on the Borel subsets of ∂Ω as follows: Note that µ Y (A) ≤ 1 and µ Y (A) ≤ 1 for all Borel A ⊂ ∂Ω, since ω and ω 1 are probability measures. Since g is Borel measurable, it suffices to show that µ Y and µ Y are Borel as claimed. Moreover, we have already observed that (5.12) holds in the special case that g is continuous on ∂Ω, thus it suffices simply to show that µ Y and µ Y are Borel measures, since equality then follows by equality on the continuous functions; in turn, it therefore suffices to show that µ Y and µ Y are countably additive on the Borel subsets of ∂Ω, i.e., that and similarly for µ Y , whenever {A k } k is a countable family of disjoint Borel subsets of ∂Ω.
To this end, given such a collection {A k } k , set A := ∪ k A k , and define u(X) := ω X (A) , u k (X) := ω X (A k ) .
Since harmonic measure is a probability measure, and in particular is countably additive, we then have (5.14) Recall that harmonic measure and surface measure are mutually absolutely continuous on the boundary of a chord-arc domain. Consequently, by (5.14), Lemma 5.4 (applied in the bounded chord-arc domain Ω 1 ), and monotone convergence, we find that The argument to treat µ Y is similar but simpler, requiring only countable additivity of harmonic measure in lieu of Lemma 5.4, and we omit the details.
6. Proof of Theorem 1.1 We now move to the proof of Theorem 1.1. Although we can still follow the original strategy of Varopoulos [Var78], consisting of ε-approximation and iteration, we have to be more careful with our construction. For example, the ε-approximators in our setting may not have pointwise non-tangential boundary traces but rather only one-sided traces in the sense of Lemma 4.14 (see Remark 4.15). We shall therefore rely on the construction of an unambiguously defined (at least 1-sided) non-tangential trace, as outlined in Remark 4.16. In addition, absolute continuity of harmonic measure with respect to surface measure may fail in the present generality, but Lemma 5.6 will allow us to make harmonic extensions, and to relate the non-tangential traces of these extensions to the data, thus allowing us to follow the basic strategy of Varopoulos.
In this section, Ω ⊂ R n+1 is an open set with n-UR boundary ∂Ω.
Suppose that f is a Borel measurable function on ∂Ω, with f L ∞ (∂Ω,σ) < ∞. We will now construct the extension Φ in Theorem 1.1.
Since f ∈ L ∞ (∂Ω, σ), there is a set Z ∈ ∂Ω, with σ(Z) = 0, such that Since σ is a Borel regular measure, there is a Borel set Z 0 ⊃ Z, with σ(Z 0 ) = 0. Set Note that f 0 = f at σ-a.e. point on ∂Ω. Moreover, f 0 is an everywhere bounded, Borel measurable function on ∂Ω, so by [Hel14, Theorem 3.9.1], we know that u 0 : Ω → R, defined by is a harmonic function in Ω satisfying where ω X is the harmonic measure on ∂Ω with pole at X. Thus, by Theorem 3.1, Lemma 3.2 and (5.5), there exists a smooth 1 2 -approximator of u 0 , i.e. a function Φ 0 ∈ C ∞ (Ω) such that where C 0 depends only on dimension and the ADR and UR constants for ∂Ω. By Lemma 4.14 and Remark 4.16, Φ 0 has a non-tangential trace (in at least a 1-sided sense), defined σ-a.e. on ∂Ω, that we denote by ϕ 0 . Furthermore, by Lemma 5.6, the non-tangential trace Tu 0 (x) exists, with , for σ-a.e. x ∈ ∂Ω . (6.1) Let Z 1 ⊂ ∂Ω denote the set where either ϕ 0 does not exist, or where (6.1) fails, hence σ(Z 1 ) = 0. Since σ is a Borel regular measure, we may assume without loss of generality that Z 1 is a Borel set. We now define Then f 1 is an everywhere bounded Borel measurable function on ∂Ω, so there is a harmonic function Again using Theorem 3.1 and Lemma 3.2, we may construct a smooth 1 2 -approximator of u 1 , i.e. a function Φ 1 ∈ C ∞ (Ω) such that with C 0 as above. By Lemma 4.14 and Remark 4.16, Φ 1 has a non-tangential trace (in at least a 1-sided sense), defined σ-a.e. on ∂Ω, that we denote by ϕ 1 . Moreover, by Lemma 5.6, u 1 has a non-tangential trace Tu 1 such that Let Z 2 ⊂ ∂Ω be the set of σ-measure 0 such that either (6.2) fails, or ϕ 1 does not exist. Again, without loss of generality, we may assume that Z 2 is a Borel set. We set We let u 2 be the harmonic extension of f 2 , and iterate, to obtain for each k ∈ N 0 , a sequence of Borel sets Z k ⊂ ∂Ω of σ-measure 0, harmonic functions u k , their 1 2 -approximators Φ k , the non-tangential boundary traces ϕ k of the approximators, and the non-tangential boundary traces f k+1 of the function u k − Φ k . These satisfy , (iii) and the triangle inequality).
On the other hand, so if harmonic measure has positive mass on Z, then v 1 = v 0 .

Carleson boxes, Carleson tents and Whitney regions
Before we prove Proposition 1.3, we revisit the construction of Whitney regions and Carleson boxes. The previous construction (see Subsection 4.1, and [HMM16, Section 3]) is not suitable for our current purposes, since the overlap of the Whitney and Carleson regions causes technical difficulties related to the Carleson measure estimates.
Since we do not need many of the strong geometric properties of the Carleson boxes constructed in [HMM16], we start by presenting a simplified construction of the boxes and proving that the boundaries of the boxes inside Ω are upper n-ADR. We note that the original proof for the upper n-ADR property of the boundaries of Carleson boxes in [HMM16, Appendix] does not apply "off-the-shelf" in our situation because we do not use dilated (hence overlapping) Whitney cubes (as is done in [HMM16,Appendix]). However, our approach makes the proof quite simple.
In this section, Ω ⊂ R n+1 is an open set, satisfying the corkscrew condition, with d-ADR boundary ∂Ω for some d ∈ (0, n], and D is a dyadic system on ∂Ω. Recall the Whitney decomposition and the definition of the collections W Q = W Q (η, K) from Subsection 4.1.
Remark 7.1. In this and the next two sections, it will be technically convenient to work with "half-open" Whitney cubes, that is, in Sections 7, 8, and 9, a cube I ∈ W is assumed to be of the form I = Π n+1 k=1 (a k , a k + h], with (I) = h ≈ dist(I, ∂Ω). All other properties of the Whitney cubes will be exactly as before.
We start by noting that our Whitney regions are not empty: Lemma 7.2. We can choose the parameters η and K depending only on the corkscrew constants, so that W Q = ∅ for every Q ∈ D.
The proof is a straightforward generalization of [HMM16, Remark 3.3] and [HM14, Lemma 5.3]. We omit the details.
Let us remark that in the codimension 1 case, if Ω = R n+1 \ E, with E n-ADR, then the corkscrew condition holds automatically, with constants that in turn depend only on dimension and ADR. Moreover, in the d-ADR case with d < n, Ω = R n+1 \ E has only one connected component, which necessarily satisfies the corkscrew condition. Remark 7.4. We note that every I ∈ W with (I) diam(∂Ω) belongs to the collection W Q I , where as above (Q I ) = (I) ≈ dist(I, Q I ), and Q I is chosen to minimize dist(I, Q I ).
Moreover, for η chosen small enough and K large enough depending only on the properties of the Whitney decomposition, every J ∈ W whose closure touches the closure of I, also belongs to W Q I . Consequently, for such η and K, we have: Remark 7.5. Given m ∈ (1, ∞), one may choose η small enough and K large enough, depending on m, so that the dyadic cone Γ(x) contains (at least locally) a cone of the type We omit the routine proof of this fact.
We now fix a suitably large aperture constant m that allows us to apply Lemma 3.14 later. Combining Lemma 7.2 and Remarks 7.4 and 7.5, we see that we may (and do) choose η and K depending only on the corkscrew constants, the Whitney cube constants, and the fixed aperture parameter m, in such a way that the collections W Q are non-empty, the Carleson boxes T Q have good covering properties and the dyadic cones contain "regular" cones. The sets U Q , T Q and Γ(x) then satisfy the same properties (with possibly different implicit constants) as U Q , T Q and Υ Q (x) in Lemma 4.12, excluding naturally the last two properties related to the bilateral corona decomposition.
Next we prove that the boundaries of the boxes T Q in Ω are upper n-ADR. The boundaries of the boxes constructed in [HMM16] are also lower n-ADR, but for our present purposes we shall need only the upper n-ADR property. We first prove a preliminary lemma, which will also be useful in the sequel.
Lemma 7.6. Let Q ∈ D. Then for each positive κ < ∞ (7.7) Proof. Note that the number of Whitney cubes in W Q is uniformly bounded for each Q , and that for I ∈ W Q we have H n (∂I) ≈ (Q ) n , by the definition of W Q ; consequently Organizing the subcubes of Q by dyadic generation we obtain by the thin boundary property (Theorem 2.16 (v)) that Combining these observations, we obtain in the codimension 1 case d = n that Lemma 7.9. For each Q, the set ∂T Q is upper n-ADR, where ∂T Q := ∂T Q ∩ Ω: for every X ∈ ∂T Q and every R ∈ (0, diam(T Q )) we have where the implicit constant depends only on n, the ADR constant, the corkscrew constant, the Whitney constants, and the fixed aperture parameter m.
Proof. Note that if X ∈ ∂T Q , then by construction there exists a dyadic cube Q ∈ D Q and a Whitney cube I ∈ W Q such that X ∈ ∂I. Also, if (Q ) (Q) and dist(Q , Q c ) (Q ) for Q ∈ D Q , then ∂I ∩ ∂T Q = ∅ for every I ∈ W Q . Thus, if I ⊂ T Q , with ∂I ∩ ∂T Q = ∅, then I ∈ W Q for a cube Q ∈ D Q such that dist(Q , Q c ) (Q ), where the implicit constant depend on η and K (which, in turn, we have chosen to depend only on the corkscrew constants, the Whitney constants, and m).
Consequently, using Lemma 7.6, we obtain Thus, we have H n ( ∂T Q ) (Q) n ≈ diam(Q) n for any Q ∈ D. Let us then prove the upper n-ADR property. Suppose that X ∈ ∂T Q and R ∈ (0, diam(T Q )). There are three cases: 1) Suppose that R ≈ diam(T Q ). Then, by the consideration above, we have 2) Suppose that R δ(X). Then, by construction, B(X, R) ∩ ∂T Q is contained in a union of a uniformly bounded number of boundaries of Whitney cubes I such that (I) > R. Since ∂I is clearly n-ADR for each I ∈ W, we therefore find that H n ( ∂T Q ∩ B(X, R)) R n . 3) Suppose that δ(X) R diam(T Q ). Then ∂T Q ∩ B(X, R) = ∂T Q ∩ B(X, R) for some subcube of Q ∈ D Q with (Q ) ≈ R. Thus, by the consideration above, we have This completes the proof.

Modified Carleson tents
Fix a cube Q 0 ∈ D. For all Q ⊆ Q 0 , we shall now construct disjoint Carleson tents t Q , that have better covering properties than τ Q . We let {Q 0 } be "generation zero", and then enumerate the dyadic descendants of Q 0 : let {Q i 1 } i be the first generation of descendants, {Q i 2 } i the second generation of descendants, and so on. Let the number of descendants of generation k be N (k). We construct a restricted version of the Whitney collection W Q , Q ⊂ Q 0 , by removing some of the cubes from W Q i k : for each k, i ∈ N, i ≤ N (k), we set where of course the second union is vacuous if i = 1, and both are vacuous if k = 0. Note that the restricted Whitney collections {W r Q } Q⊂Q 0 are pairwise disjoint, by construction. We can then define restricted Whitney regions U r Q and modified Carleson tents t Q for cubes Q ⊆ Q 0 : . Two modified Carleson tents t Q and t Q in the simplest case where Ω = R 2 + . The boundary they share may be slightly messy but it consists of a union of faces of Whitney cubes.
Remark 8.2. Since the Whitney collections {W r Q } Q⊂Q 0 are pairwise disjoint, and since we are now working with half-open (hence disjoint) Whitney cubes I, it follows that the sets {U r Q } Q⊂Q 0 are also pairwise disjoint. Lemma 8.3. Suppose that Q, Q 1 , Q 2 ∈ D Q 0 . We then have: , of uniformly bounded cardinality N depending only on n, ADR, η and K, such that Proof. The properties ii), iii), and iv) follow directly from the construction so we prove only property i).
Note that by construction (see Definition 7.3), and that U r Q ⊂ U Q for every Q ∈ D Q 0 . Moreover, the restricted Whitney regions U r Q are disjoint (see Remark 8.2). Consequently, Lemma 8.4. The sets ∂t Q ∩ Ω are upper n-ADR with the ADR constant depending only on the dimension and the ADR constant of ∂Ω.
Proof. Recall that τ Q ⊂ t Q , by Lemma 8.3 i). Thus, if I ⊂ t Q , with ∂I ∩ ∂t Q = ∅, then I ∈ W r Q for a cube Q ∈ D Q such that dist(Q , Q c ) (Q ). One may then use Lemma 7.6, following the proof of Lemma 7.9 with minor adjustments. We omit the details. 9. Proof of Proposition 1.3 Suppose that Ω ⊂ R n+1 is an open set satisfying the corkscrew condition with d-ADR boundary for some d ∈ (0, n]. Let Q 0 ∈ D be a fixed dyadic cube, D Q 0 = {Q j } j ⊂ D Q 0 be a collection of subcubes of Q 0 and {α j } j a collection of coefficients such that belongs to BMO(∂Ω), the collection D Q 0 enjoys a Carleson packing condition with packing norm C D Q 0 =: C 0 (see Definition 2.23), and sup j |α j | f BMO . Note that f vanishes on ∂Ω \ Q 0 , but we assume that f ∈ BMO, globally on ∂Ω. We denote where t Q j is the modified Carleson tent defined in (8.1). We will show that a smooth version of F 0 satisfies the properties in Proposition 1.3.
We start by proving the following estimate that we shall need later: Lemma 9.1. Let Q, Q ∈ D be such that where the implicit constant depends on the implicit constant in (9.2).
Proof. Let us fix two disjoint cubes Q, Q ∈ D, that satisfy (9.2). Fix a constant C large enough (depending only on the implicit constants in (9.2)) that Q ∪ Q ⊂ B * Q := B(x Q , r), with r := C (Q). Let ∆ * Q := B * Q ∩ ∂Ω denote the corresponding surface ball. Since f ∈ BMO(∂Ω), by the ADR property we have By the uniform bound on the coefficients and the packing condition of the collection {Q j } j , we have that and similarly with Q in place of Q. Combining this observation with (9.3), we see that and similarly By the triangle inequality, these last two estimates yield Lemma 9.4. We have for σ-a.e. x ∈ ∂Ω. Here lim Y →x N.T. stands for standard type non-tangential convergence.
Proof. By the Carleson packing condition of D Q 0 , and the uniform boundedness of the coefficients α j , it follows that j 1 Q j (x) < ∞, and hence also | j α j 1 Q j (x)| < ∞, for σ-a.e. x ∈ ∂Ω. Also, j α j 1 t Q j (Y ) < ∞ for each Y ∈ Ω, since Y can belong to only a finite number of modified tents t Q j (those for which (Q 0 ) ≥ (Q j ) δ(Y )). Thus, is absolutely convergent for σ-a.e. x ∈ ∂Ω, and all Y ∈ Ω. For fixed x with j 1 Q j (x) < ∞, we split In particular, for x ∈ ∂Ω \ Q 0 , we have F 2 (x) = D Q 0 , and F 2 0 (x, Y ) = F 0 (x, Y ), since Q j ⊂ Q 0 for each j.
Let us then show that lim Y →x N.T. F i 0 (x, Y ) = 0, i = 1, 2, for almost every x. Suppose that ε > 0, Y ε ∈ Γ(x) and dist(x, Y ε ) < ε. For those j such that x ∈ Q j , we have where sup j |α j | f BMO by assumption. Recall that we have fixed x with j 1 Q j (x) < ∞.
is the tail of a convergent series, so that I ε 1 → 0 as ε → 0. Turning now to I ε 2 , we first note that since Y ε ∈ Γ(x)\t Q j , there exists a cube Q x, such that Y ε ∈ U Q \ t Q j , with (Q) ≈ δ(Y ε ) ε for some uniformly bounded implicit constants. If ε is small enough, then (Q) (Q j ) and thus Q ⊂ Q j , since x ∈ Q ∩ Q j . Hence also t Q ⊂ t Q j . Consequently, Y ε / ∈ t Q , and therefore there exists another cube Q such that (Q ) ≈ (Q), Y ε ∈ t Q , and Q ∩ Q j = ∅. In particular, dist(x, Q c j ) ε ≤ √ ε (Q j ). We set for the same implicit uniform constant as above, and assume that ε is so small that this constant times √ ε is a lot smaller that 1. We then have In particular, by (2.18) and the Carleson packing condition of {Q j } j we obtain Thus, there is a sequence (ε k ) k with ε k → 0 as k → ∞, such that h ε k (x) → 0 as k → ∞, for σ-a.e. x ∈ ∂Ω. Since h ε is pointwise decreasing as ε 0, we therefore have h ε (x) → 0 as ε → 0, and hence also lim ε→0 I ε 2 (x) = 0, for σ-a.e. x ∈ ∂Ω. Consider now those j such that x / ∈ Q j . Then where, as before, sup j |α j | f BMO by assumption.
Consequently, by the triangle inequality, there exists a uniformly bounded constant c ≥ 1 such that x ∈ cQ j (recall Notation 2.20 (4)). Thus, we have J ε 1 (x) ≤ j: (Q j )< √ ε 1 cQ j (x). By the Carleson packing condition of {Q j } j , we know that j 1 cQ j (x) < ∞ for almost every x. Therefore J ε 1 (x) is bounded by the tail of a convergent series for almost every x, hence J ε 1 (x) → 0 as ε → 0 for almost every x. For J ε 2 , we can use similar but simpler arguments as with I ε 2 in the previous case. Since Y ε ∈ Γ(x) ∩ t Q j , there exists a subcube Q ⊂ Q j such that (Q) ≈ δ(Y ε ) ε and Y ε ∈ U Q . By definition, we have Y ε ∈ Γ(y) and dist(y, Y ε ) δ(Y ε ) for every y ∈ Q. In particular, there exists a point y ∈ Q j such that dist(x, Q j ) ≤ dist(x, y) ε ≤ √ ε (Q j ). We now set and proceed as we did for I ε 2 , but now using the exterior thin boundary estimate (2.22) in lieu of (2.18). We leave the remaining details to the reader.
Remark 9.5. The previous lemma is true also if we define the extension F 0 with respect to the overlapping boxes T Q or the tents t Q and in those cases the proof actually becomes simpler. However, in the next proof it is crucial that we use the modified Carleson tents.
Lemma 9.6. The measure |∇F 0 (Y )| dY satisfies a quantitative codimension 1 type Carleson measure estimate: It is easy to see that every ball B(x, R) ∩ Ω with x ∈ ∂Ω and R diam(∂Ω) can be covered by the union of interiors of a uniformly bounded number of Carleson boxes T Q , with R ≈ (Q) (see [HMM16, p. 2353[HMM16, p. -2354 for details). Thus, it is enough to show thaẗ for an arbitrary cube Q ∈ D. We consider first the case that Q ⊂ Q 0 . Fix Q ⊂ Q 0 and a vector field − → Ψ ∈ C 1 0 (int(T Q )) such that − → Ψ L ∞ ≤ 1. We havë where M is a sufficiently large positive integer to be chosen. The sum J 1 is easy. Since ∂t Q j ∩ Ω is a union of faces of Whitney cubes, and the support of − → Ψ has a strictly positive distance to ∂Ω, we can apply the divergence theorem to geẗ where in the last step we have used Lemma 8.4. Since T Q contains the support of − → Ψ , every Q j appearing in J 1 is contained in a ball B * * Q := B(x Q , C (Q)), for some C chosen large enough depending on M , η and K. Combining the latter fact with (9.7), and using the Carleson packing condition for the collection {Q j } (and Lemma 2.24 in the higher codimension case d < n), we see that J 1 C 0 (Q) n .
The sum J 2 is little trickier. Since − → Ψ is compactly supported in int(T Q ), we have − → Ψ = 0 on ∂T Q . In particular, if we happen to have T Q = t Q , then T Q ∩ t Q j = T Q for every Q j in the sum J 2 , and the same divergence theorem argument as above implies that J 2 = 0. Unfortunately, usually t Q T Q , so we have to be more careful.
By Lemma 8.3, there is a collection F(Q) = {Q i } N i=1 , of uniformly bounded cardinality N , with (Q i ) = (Q i ) ≈ η,K (Q) for each i, i , such that ∪ i t Q i contains T Q . We now choose M = M (η, K) so that (Q i ) = 2 M (Q), for every Q i ∈ F(Q). Thus, the cubes Q j in J 2 satisfy Q i ∩ Q j ∈ {∅, Q i } for all i and j. This choice and the divergence theorem give where we have used Lemma 8.3 ii) and iii). Since supp( − → Ψ ) ⊂ int(T Q ), − → Ψ (X) can be non-zero only if X lies in the interior of T Q . Furthermore, the modified Carleson tents t Q i are disjoint, and their union covers T Q . Thus, for every point X on ∂t Q i where − → Ψ (X) is non-zero, there is a different cube Q k ∈ F(Q) such that X ∈ ∂t Q k .
By Lemma 8.3, we have that ∂t Q i ∩ ∂t Q k ∩ Ω is either empty or it consists of a union of faces of Whitney cubes. Let us define the set of all the pairs of indices of the cubes Q i by setting P := {(i, k) : 1 ≤ i < k ≤ N } and let us define the collection of the faces of Whitney cubes between t Q i and t Q k by setting F (i,k) := {F : F is a face of a Whitney cube contained in ∂t Q i ∩ ∂t Q k } for every (i, k) ∈ P. Notice that F (i,k) may be empty. We can now write where − → N i is the outer unit normal of ∂t Q i ∩ Ω. We notice that on F the normals − → N i and − → N k point to the opposite directions. Thus, we actually have By Lemma 9.1, we therefore have Furthermore, if F ∈ F (i,k) , then by definition F ⊂ ∂t Q i ∩ ∂t Q k , so since ∂t Q i ∩ Ω is upper n-ADR by Lemma 8.4, and (Q) ≈ (Q i ). The number of the modified Carleson tents t Q i was uniformly bounded, hence, so is the cardinality of the set P. Thus, J 2 C 0 f BMO (Q) n . This completes the proof in the case Q ⊂ Q 0 .
by Lebesgue's differentiation theorem, where the latter identity is valid for σ-a.e. x such that N f (x) is infinite. Setting we obtain the claimed decomposition in (2). It remains to check that with this definition, we have f L ∞ (E,dσ) f BMO D . To this end, observe that in order to have N f (x) < ∞, we must have that for every dyadic cube Q with x ∈ Q Q min (x), otherwise, there would have been another stopping cube containing x, and strictly contained in Q min (x), which contradicts the definition of Q min (x). By Lebesgue's differentiation theorem, we therefore find that | f (x)| ≤ 2 f BMO D for σ-a.e. x such that N f (x) < ∞, so that (2) holds.
(3) By a standard limiting argument, we may assume that the collection D Q 0 is finite. We first notice that by the stopping conditions we have σ(Q where I 1 is the sum over those Q (i) j such that P (i−1) k(j) ⊂ Q and I 2 is the sum over the rest of the relevant cubes. The cubes in the sum I 2 are disjoint and thus, I 2 ≤ σ(Q). Let i(Q) be the smallest integer such that F i(Q) contains at least one cube in the sum I 1 ; thus, I 2 is the sum over the cubes in F i(Q)−1 that are contained in Q. With this notation, we may write I = i≥i(Q)−1 R∈F i ,R⊂Q σ(R) = i≥i(Q) R∈F i ,R⊂Q σ(R) + I 2 = I 1 + I 2 .
By Theorem 1.1, we know that there exists a function Φ ∈ C ∞ (Ω) such that Φ converges to f non-tangentially almost everywhere, the measure µ 1 := |∇Φ(Y )| dY is a Carleson measure and C µ 1 f L ∞ (∂Ω) f BMO(∂Ω) . By the decomposition f = f + f 0 , we know that f 0 is a BMO function as it is a sum of two BMO functions. Thus, by Proposition 1.3, there exists a function F ∈ C ∞ (Ω) such that F converges to f 0 non-tangentially almost everywhere, the measure µ 2 := |∇F (Y )| dY is a Carleson measure and Thus, we can set V := Φ + F .