Uniform rectifiability implies Varopoulos extensions

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Uniform rectifiability implies Varopoulos extensions © 2021 the Authors Published version Hofmann, Steve; Tapiola, Olli Hofmann, S., & Tapiola, O. (2021). Uniform rectifiability implies Varopoulos extensions. Advances in Mathematics, 390, Article 107961. https://doi.org/10.1016/j.aim.2021.107961 2021 Advances in Mathematics 390 (2021) 107961 Contents lists available at ScienceDirect Advances in Mathematics www.elsevier.com/locate/aim Uniform rectifiability implies Varopoulos extensions ✩ Steve Hofmann a, Olli Tapiola b,∗ aDepartment of Mathematics, University of Missouri, Columbia, MO 65211, USA bDepartment of Mathematics and Statistics, P.O. Box 35 (MaD), FI-40014 University of Jyväskylä, Finland a r t i c l e i n f o a b s t r a c t Article history: Received 28 June 2020 Received in revised form 9 July 2021 Accepted 10 July 2021 Available online xxxx Communicated by C. Fefferman Keywords: Uniform rectifiability Carleson measure estimate Epsilon-approximability BMO Solvability of the Dirichlet problem Harmonic measure We construct extensions of Varopolous type for functions f∈ BMO(E), for any uniformly rectifiable set Eof codimension one. More precisely, let Ω ⊂Rn+1 be an open set satisfying the corkscrew condition, with an n-dimensional uniformly rectifiable boundary ∂Ω, and let σ:=Hn∂Ωdenote the surface measure on ∂Ω. We show that if f∈BMO(∂Ω, dσ) with compact support on ∂Ω, then there exists a smooth function Vin Ωsuch that |∇V(Y)| dY is a Carleson measure with Carleson norm controlled by the BMO norm of f, and such that Vconverges in some non-tangential sense to falmost everywhere with respect to σ. Our results should be compared to recent geometric characterizations of Lp-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∈Cc(∂Ω), a harmonic extension u, with |∇u(Y)|2dist(Y, ∂Ω) dY a Carleson measure with Carleson norm controlled by the BMO norm of f, only in the presence of an appropriate quantitative connectivity condition. © 2021 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). ✩S.H. was supported by NSF grant DMS-1664047. O.T. was partially supported by Emil Aaltosen Säätiö through Foundations’ Post Doc Pool grant. *Corresponding author. E-mail addresses: [email protected] (S. Hofmann), [email protected] (O. Tapiola). https://doi.org/10.1016/j.aim.2021.107961 0001-8708/© 2021 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). 2S. Hofmann, O. Tapiola / Advances in Mathematics 390 (2021) 107961 Contents List of symbols.......................................................... 2 1. Introduction ....................................................... 2 2. Notation and basic definitions ........................................... 6 3. ε-approximability and regularization....................................... 12 4. Bilateral corona decomposition and one-sided non-tangential traces ................. 17 5. Some results on boundary behavior of bounded harmonic functions ................. 25 6. Proof of Theorem 1.1 ................................................. 30 7. Carleson boxes, Carleson tents and Whitney regions............................ 34 8. Modified Carleson tents ............................................... 38 9. Proof of Proposition 1.3 ............................................... 40 10. Garnett’s decomposition lemma and proof of Theorem 1.2 ........................ 47 References ............................................................. 51 List of symbols CμCarleson norm of the measure μ(Definition 2.4) CACarleson packing norm of A ⊂D(Definition 2.23) Dcollection 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) ΥQ(x) semi-closed truncated cone at x ∈Q ⊂∂Ω (Section 4)  ΥQ(x)interior of ΥQ(x) Ωopen set in Rn+1 with ADR boundary ∂Ω ωXharmonic measure with pole at X∈Ω UQ,U Qdilated and non-dilated closed Whitney region (Sections 4and 7) Ur Qclosed restricted Whitney region (Section 8) TQ,T Qsemi-closed and open Carleson box (Sections 4and 7) τQCarleson tent (Definition 7.3) tQmodified Carleson tent (Section 8) WWhitney cubes in Ω (Sections 4and 7) GQ0counting function with respect to Q0∈D(Lemma 4.13) δ, β distance and smooth distance function with respect to ∂Ω(Theorem3.3) fA,fflAfintegral average of fover A(Section 2) N0the set of non-negative integers {0, 1, 2, 3, ...} 1. 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 [36]) 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) S. Hofmann, O. Tapiola / Advances in Mathematics 390 (2021) 107961 3 boundaries (see [20]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 [41,42], 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 [4]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 [21] 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 ([4,21]). Let Ω ⊂Rn+1 be an open set satisfying the corkscrew condition and suppose that ∂Ωis n-ADR. Then the following conditions are equivalent: (1) ∂Ωis UR and Ωsatisfies the weak local John condition, (2) harmonic measure belongs to the class weak-A∞with respect to the surface measure σ:=Hn∂Ωon ∂Ω, (3) the Dirichlet problem is Lp-solvable for some p <∞, (4) the Dirichlet problem is BMO-solvable. By Lp-solvability we mean that there exists a constant Csuch that if f∈Lp(∂Ω), then the solution uto the Dirichlet problem with data fconverges non-tangentially to fand N∗uLp(∂Ω) ≤CfLp(∂Ω), where N∗is a non-tangential maximal operator. Many key results related to this concept can be found in the monograph of Kenig [35]. By BMO solvability,1we mean that there exists a constant Csuch that if fis a compactly supported continuous function on ∂Ω, then the solution uto the Dirichlet problem satisfies the Carleson measure estimate sup x∈∂Ω,0<rdiam(∂Ω) 1 σ(Δ(x, r)) ¨ Ω∩B(x,r) |∇u(Y)|2δ(Y)dY ≤Cf2 BMO(∂Ω), 1The definition is slightly different if Ωis unbounded and ∂Ωis bounded; see [21, Section 5] for details. 4S. Hofmann, O. Tapiola / Advances in Mathematics 390 (2021) 107961 where Δ(x, r) :=B(x, r) ∩∂Ω. This type of solvability was first shown to be equivalent to Lp-solvability, for some p <∞, by Dindos, Kenig and Pipher [14], in Lipschitz or chord-arc domains (see also [43]for an extension to 1-sided chord-arc domains). It was previously known that the weak-A∞property of harmonic measure (equivalently, Lp-solvability for some p <∞) may fail in the absence of connectivity, even if the boundary is UR [5], but the result of [4]is the first that tells us precisely how much connectivity we need (although we refer the reader to related work of Azzam [2], 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 Lp-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 ⊂Rn+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. ii) ΦL∞(Ω) ≤CfL∞(∂Ω), iii) limY→xN.T. Φ(Y) =f(x)for σ-a.e. x ∈∂Ω, iv) |∇Φ(Y)| dY is a Carleson measure: sup r>0,x∈∂Ω 1 rn¨ B(x,r)∩Ω |∇Φ(Y)|dY ≤CfL∞. Here, limY→xN.T. stands for one-sided non-tangential convergence2; σ:=Hn∂Ωis the surface measure, and δ(X) :=dist(X, ∂Ω) for X∈Ω. The constant Cdepends only on n, and the UR and corkscrew constants. We remark that in particular, Theorem 1.1 applies in the case that Ω :=Rn+1 \E, where Eis an arbitrary n-UR set: the corkscrew condition in that case is a simple (and well-known) consequence of Ahlfors–David regularity of E. The proof is based on a combination of geometric arguments, potential theory and dyadic analysis, but the basic strategy follows that of Varopoulos [41,42]: in particular, we strongly make use of the ε-approximability property of harmonic functions, established in the present context in [25](see Theorem 3.1 below). However, the implementation of this program is a delicate matter in the present generality, owing to the need to 2The notion of non-tangential convergence must be suitably interpreted in the present context. We shall return to this matter in the sequel; see Definition 2.1, Lemma 4.14, and Remarks 2.2, 4.15 and 4.16. S. Hofmann, O. Tapiola / Advances in Mathematics 390 (2021) 107961 5 make harmonic extensions of functions belonging to L∞(∂Ω, dσ), with non-tangential convergence σ-a.e. to the data, even though harmonic measure may fail to be absolutely continuous with respect to surface measure σ; see Sections 5and 6, and in particular Remark 6.4. Originally, the notion (although not the terminology) of ε-approximability was introduced by Varopoulos [42], and refined by Garnett [18], in order to study new ways to extend BMO functions inspired by Carleson’s corona theorem [8], and the closely related topic of H1–BMO duality (see particularly [16, Theorem 3]). The ε-approximability property provides a convenient detour to circumvent the unfortunate fact that there exist harmonic functions usuch that |∇u(Y)| dY is not a Carleson measure [18]. Subsequently, this property has offered ways to connect Carleson measure estimates for solutions, with quantitative Fatou Theorems [18], [6], with absolute continuity properties of elliptic measures [34,27]and with boundary geometry [25,17,3,6,23,7]. Our second result is the following generalization of [42, Theorem 2], which in some sense provides a substitute for BMO-solvability of the Dirichlet problem: Theorem 1.2. Suppose that Ω ⊂Rn+1 is an open set satisfying the corkscrew condition with n-UR boundary. Then there exists a constant Csuch that if f∈BMO(∂Ω, dσ)is compactly supported, then there exists a function V=Vfin Ωsuch that i) V∈C∞(Ω), and |∇V(X)| ≤CfBMO δ(X)−1, for all X∈Ω, ii) limY→xN.T. V(Y) =f(x)for σ-a.e. x ∈∂Ω, iii) |∇V(Y)| dY is a Carleson measure: sup r>0,x∈∂Ω 1 rn¨ B(x,r)∩Ω |∇V(Y)|dY ≤CfBMO. Here limY→xN.T. stands for one-sided non-tangential convergence (see Definition 2.1 and Remark 2.2); δ(X) :=dist(X, ∂Ω), and σ:=Hn∂Ωis the surface measure. The proof is a combination of Theorem 1.1, Garnett’s decomposition lemma (see Lemma 10.1), and the following extension result for the “dyadic part” of Garnett’s lemma: Proposition 1.3. Suppose that Ω ⊂Rn+1 is an open set satisfying the corkscrew condition with d-ADR boundary for some d ∈(0, n]. Let Dbe a dyadic system on ∂Ω, Q0∈D be a fixed dyadic cube and {Qj}j⊂DQ0be a collection of subcubes of Q0. Suppose that function fin ∂Ω, f(x) =jαj1Qj, satisfies the following conditions for some C0≥1: •f∈BMO(∂Ω), •Qj⊂Qσ(Qj) ≤C0σ(Q)for every Q ∈D, •supj|αj| ≤cfBMO, 6S. Hofmann, O. Tapiola / Advances in Mathematics 390 (2021) 107961 where c ≥1depends only on the dimension and the ADR constant (see Definition 2.11). Then there exists a function F=Ffin Ωsuch that i) F∈C∞(Ω), and |∇F(X)| ≤C1fBMO δ(X)−1, for all X∈Ω, ii) limY→xN.T. F(Y) =f(x)for σ-a.e. x ∈∂Ω, iii) |∇F(Y)| dY satisfies a quantitative codimension 1type Carleson measure estimate: sup r>0,x∈∂Ω 1 rn¨ B(x,r)∩Ω |∇F(Y)|dY ≤C1C0fBMO (1.4) for a constant C1≥1depending only on the dimension and the ADR constant. Here limY→xN.T. stands for standard type non-tangential convergence (see Definition 2.1); δ(X) :=dist(X, ∂Ω), and σ:=Hd∂Ωis the surface measure. 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. 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 4and 5, and we prove the theorem in Section 6. In Sections 7and 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. Acknowledgments The authors wish to thank the anonymous referee for many insightful comments related to this work and related questions, and for several suggestions that have helped to clarify and improve the presentation. 2. Notation and basic definitions We use the following notation. S. Hofmann, O. Tapiola / Advances in Mathematics 390 (2021) 107961 7 •Ω ⊂Rn+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 ∂Ωis n-UR (see Definition 2.13) and that Ωsatisfies the corkscrew condition (see Definition 2.12). •The letters cand Cdenote 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 cand Cmay change from one occurrence to another. We do not track how our bounds depend on these constants and usually just write γ1γ2if γ1≤cγ2for a constant like this cand γ1≈γ2if γ1γ2γ1. If the constant cκ depends only on parameters of the previous type and some other parameter κ, we usually write γ1κγ2instead of γ1≤cκγ2. •We use capital letters X, Y, Z, and so on to denote points in Ωand lowercase letters x, y, z, and so on to denote points in ∂Ω. •The (n + 1)-dimensional Euclidean open ball of radius rwill be denoted B(x, r) or B(X, r) depending on whether the center point lies on ∂Ωor Ω. We denote the surface ball of radius rcentered at xby Δ(x, r) :=B(x, r) ∩∂Ω. •Given a Euclidean ball B:=B(X, r)or a surface ball Δ :=Δ(x, r)and constant κ >0, we denote κB :=B(X, κr)and κΔ :=Δ(x, κr). •For every X∈Ωwe set δ(X) :=dist(X, ∂Ω). •We let Hdbe the d-dimensional Hausdorff measure and denote the surface measure of ∂Ωby σ:=Hd∂Ω. The (n + 1)-dimensional Lebesgue measure of a measurable set A ⊂Ω will be denoted by |A|. •For a set A ⊂Rn+1, we let 1Abe the indicator function of A: 1A(x) =0if x /∈A and 1A(x) =1if x ∈A. •The interior of a set Awill be denoted int(A). •The unit outer normal (when it exists) will be denoted by −→ N. •For μ-measurable sets Awith positive and finite measure we set fA:=fflAfdμ := 1 μ(A)fdμ. Definition 2.1 (Cones and non-tangential limits). Suppose that m >1. For every x ∈∂Ω, the cone of m-aperture at xis the set  Γ(x):= Γm(x):={Z∈Ω: dist(Z, x)<mδ(Z)}. Let Gbe a function defined in Ω, gbe a function defined on ∂Ωand xbe a point on ∂Ω. We consider two types of non-tangential convergence in this paper. We use the notation limY→xN.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 limk→∞ G(Yk) =g(x)for every sequence (Yk)in  Γm(x)such that limk→∞ Yk=x. 8S. Hofmann, O. Tapiola / Advances in Mathematics 390 (2021) 107961 •With one-sided non-tangential convergence we mean that there exists m >1and a connected component A ⊂ Γm(x)such that x ∈∂A and limk→∞ G(Yk) =g(x) for every sequence (Yk)in Asuch that limk→∞ Yk=x. Remark 2.2. i) If Ω ⊂Rn+1 is an open set satisfying the corkscrew condition, with UR boundary ∂Ω, then for σ-a.e. x ∈∂Ω, the cone with vertex at xhas at most two connected components inside Ωsuch 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 4and 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 fsuch that fBMO :=sup Δ Δ |f(y)−fΔ|dσ(y)<∞, where the supremum is taken over all surface balls Δ ⊂∂Ω. We define the dyadic BMO space BMOD(∂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 μin Ωis a Carleson measure (with respect to ∂Ω)if we have Cμ:=sup x∈∂Ω,r>0 μ(B(x, r)∩Ω) rn<∞.(2.5) We call Cμthe Carleson norm of μ. Definition 2.6 (Local BV). We say that locally integrable function fhas locally bounded variation in Ω (denote f∈BVloc(Ω)) if for any open relatively compact set Ω⊂Ωthe total variation over Ωis finite: ¨ Ω|∇ϕ|dY :=sup −→ Ψ∈C1 0(Ω) −→ ΨL∞(Ω)≤1 ¨ Ω ϕdiv−→ ΨdY < ∞, where C1 0(Ω)is the class of compactly supported continuously differentiable vector fields in Ω. S. Hofmann, O. Tapiola / Advances in Mathematics 390 (2021) 107961 15 δ(X)−n−1¨ BX |∇G0(Y)|dY , (3.12) where we have used also (3.4), and Poincaré’s inequality for BV (see [15, Theorem 1, p. 189]). Now let ˆx∈∂Ωbe a “touching point” for X, i.e. |X−ˆx| =δ(X). Then δ(X)−n−1¨ BX |∇G0(Y)|dY δ(X)−n−1¨ B(ˆx,2δ(X))∩Ω |∇G0(Y)|dY Cμ δ(X), 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 δ(X)−n¨ BX |∇G0(Y)|dY ≤C.  Lemma 3.13. If G0∈BVloc(Ω) and μ =|∇G0(Y)| dY is a Carleson measure, then also |∇G(Y)| dY is a Carleson measure and sup r>0,z∈∂Ω 1 rn¨ B(z,r)∩Ω |∇G(X)|dX Cμ where Cμis the constant in (2.5). Proof. Fix B(z, r)with z∈∂Ω. We cover B(z, r) ∩Ωby (possibly disconnected) “halfopen” regions Vk:=X∈Ω∩B(z,r):2 −k−1r≤δ(X)<2−kr, so that Ω ∩B(z, r) =∪∞ k=0Vk. Observe that for X∈Vk, the ball BXdefined in (3.4)is contained in V∗ k:=Y∈Ω∩B(z,2r):2 −k−2r≤δ(Y)<2−k+1r, and moreover, that for Y∈BX, we have |X−Y|≤δ(X)/2≈δ(Y). Thus, using (3.12), we see that 16 S. Hofmann, O. Tapiola / Advances in Mathematics 390 (2021) 107961 ¨ Vk |∇G(X)|dX ¨ Vk δ(X)−n−1¨ BX |∇G0(Y)|dY dX ¨ V∗ k |∇G0(Y)|⎛ ⎜ ⎝δ(Y)−n−1¨ |Y−X|δ(Y) dX⎞ ⎟ ⎠dY ≈¨ V∗ k |∇G0(Y)|dY. Summing in k, and using that the sets V∗ khave bounded overlaps, we obtain ¨ B(z,r)∩Ω |∇G(X)|dX ¨ B(z,2r)∩Ω |∇G0(Y)|dY Cμrn, as desired.  Lemma 3.14. If G0converges to g(x)non-tangentially in the standard (respectively, onesided) sense in a cone with large enough aperture, then also Gconverges to g(x)nontangentially in the standard (respectively, one-sided) sense. Proof. Suppose that Y∈ Γm(x)for some m >1. We recall that G(Y)= 1 β(Y)n+1 ¨ B(Y,β(Y) 2m2) ζY−Z β(Y)G0(Z)dZ. In particular, since dist(x, Y) <mδ(Y), we have dist(x, Z)≤dist(x, Y )+dist(Y,Z)<mδ(Y)+β(Y) 2m2≤m+1 2δ(Y) for every Z∈B(Y, β(Y)/2m2). Also, if |G0(Z) −g(x)| <εfor every Z∈B(Y, β(Y) 2m2), we can use the facts that ˜ζ=1and ζ(X) ≤1for every X∈Ωto show that |G(Y)−g(x)|≤ 1 β(Y)n+1 ¨ B(Y,β(Y) 2m2)ζY−Z β(Y)|G0(Z)−g(x)|dZ ε. By combining these two observations we see that if G0converges to g(x) non-tangentially in a cone with aperture m, then Gconverges 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 2m2) S. Hofmann, O. Tapiola / Advances in Mathematics 390 (2021) 107961 17 and we use dyadic cones that we can construct in such a way that they contain cones of the type  Γmfor a large m(see Section 7). Proof of Lemma 3.2.By Lemma 3.8, Lemma 3.13 and the Mean value theorem, the previous regularization method gives us the desired properties i), ii) and iii), but it is not clear if the regularized ε-approximators are still ε-approximators in the sense of Theorem 3.1. Because of this, we tweak the regularization method for ε-approximators. If Φ0is the original ε-approximator, we define the regularized ε-approximator with the formula Φ(X):=¨Λε(X,Y )Φ0(Y)dY , where Λε(X,Y ):=ζεβ(X)(X−Y). This changes the outcome of Lemma 3.8 for Φin the sense that we get a bound |∇Φ(X)|Cε εn+1δ(X) instead, where Cεis the Carleson norm of the measure |∇Φ0(Y)| dY . However, since the size of the Carleson norm of the ε-approximators is not important for us (i.e. it can depend very strongly on ε), we can simply absorb the constant ε−n−1to the Carleson norm of the regularized ε-approximators. Thus, if Φ0is an ε-approximator, then Φis a Cε-approximator such that Carleson norm  Cεof the measure |∇Φ(Y)| dY satisfies  Cε≤CCε εn+1 for some uniformly bounded constant C. We omit the details.  4. Bilateral corona decomposition and one-sided non-tangential traces In Rn+1 +, the construction of dyadic Carleson boxes and dyadic Whitney regions is very simple: just take a dyadic cube on Rn, 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. [27,30]). However, it is still possible to construct substitutes for these boxes and regions that share many good properties with their Rn+1 +-analogues [25, Section 3]. In this paper, we need two versions of the Whitney regions from [25]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. 18 S. Hofmann, O. Tapiola / Advances in Mathematics 390 (2021) 107961 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 7and 8. Let us start by recalling some key tools from [25]. In this section, Ω ⊂Rn+1 is an open set with n-UR boundary ∂Ωand Dis 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}Iis a Whitney decomposition of Ω(see e.g. [39, Chapter VI], that is, {I}Iis a collection of closed (n + 1)-dimensional Euclidean cubes whose interiors are disjoint such that II=Ωand 4diam(I)≤dist(4I,∂Ω) ≤dist(I,∂Ω) ≤40diam(I) for every I∈W (here, by 4Iwe mean the standard concentric Euclidean dilate, as opposed to the “dilate” of a dyadic cube Q ∈D(∂Ω) defined in Notation 2.20 (4)), and 1 4diam(I1)≤diam(I2)≤4diam(I1) whenever I1∩I2=∅. For parameters ηand Ksatisfying η1 Kand for every Q ∈D(∂Ω) we set W0 Q:=W0 Q(η,K):={I∈W:η1/4(Q)≤(I)≤K1/2(Q),dist(I,Q)≤K1/2(Q)}. (4.1) Remark 4.2. We note that W0 Qis non-empty, for ηchosen small enough, and Klarge enough, provided that Ωsatisfies the corkscrew condition (see [25, Section 3]). In particular, the latter is true when Ω =Ω E:=Rn+1 \E, where E⊂Rn+1 is an n-ADR set. In the sequel, we shall always assume that ηand Khave been so chosen. Definition 4.3. For ξ>1and every I∈W, we let I∗be the concentric dilation of I: I∗=I∗(ξ):=ξI. We note that if ξis 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 ξas allowable. If we choose (as above) the parameters η, Kand ξin a suitable way, the collections I∈WQIand I∈WQI∗, and certain variants of these collections, have strong geometric properties that we will formulate in the next lemmas and use in the subsequent sections. S. Hofmann, O. Tapiola / Advances in Mathematics 390 (2021) 107961 19 Definition 4.4. We say that a subcollection S⊂Dis coherent if the following three conditions hold. (a) There exists a maximal element Q(S) ∈Ssuch that Q ⊂Q(S)for every Q ∈S. (b) If Q ∈Sand P∈Dis a cube such that Q ⊂P⊂Q(S), then also P∈S. (c) If Q ∈S, then either all children of Qbelong to Sor none of them do. If Ssatisfies only conditions (a) and (b), then we say that Sis semicoherent. Lemma 4.5 ([25, Lemma 2.2]). For any pair of positive constants η1and K1 there exists a disjoint decomposition D=G∪Bsatisfying the following properties: (1) The “good” collection Gis a disjoint union of coherent stopping time regimes S. (2) The “bad” collection Band the maximal cubes Q(S)satisfy a Carleson packing condition: for every Q ∈Dwe have  Q⊂Q,Q∈B σ(Q)+  S:Q(S)⊂Q σ(Q(S)) ≤Cη,Kσ(Q). (3) For every S, there exists an n-dimensional Lipschitz graph ΓS, with Lipschitz constant at most η, such that for every Q ∈Swe have sup x∈Δ∗ Q dist(x, ΓS)+ sup y∈B∗ Q∩ΓS dist(y,∂Ω) <η(Q), where B∗ Q:=B(xQ, K(Q)) and Δ∗ Q:=B∗ Q∩∂Ω. We call the decomposition D=G∪Bin Lemma 4.5 the bilateral corona decomposition of D. Next, we recall a construction in [25, Section 3], leading up to and including in particular [25, Lemma 3.24]. We summarize this construction as follows. Lemma 4.6. Let E⊂Rn+1 be UR, and set ΩE:=Rn+1 \E. Given positive constants η1and 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 W0 Qin (4.1)has an augmentation W∗ Q⊂Wsatisfying the following properties. (1) W0 Q⊂W ∗ Q=W∗,+ Q∪W∗,− Q, where (after a suitable rotation of coordinates) each I∈W∗,+ Qlies above the Lipschitz graph ΓSof Lemma 4.5, each I∈W∗,− Qlies below ΓS. 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 ΩEas each I∈W∗,+ Q(resp. I∈W∗,− Q) and W∗,+ Q∩W∗,+ Q=∅(resp. W∗,− Q∩W∗,− Q=∅). 20 S. Hofmann, O. Tapiola / Advances in Mathematics 390 (2021) 107961 (2) There are uniform constants c <1, C>1, and ξ0>1such that cη1/2(Q)≤(I)≤CK1/2(Q),∀I∈W∗ Q, dist(I,Q)≤CK1/2(Q),∀I∈W∗ Q, cη1/2(Q)≤dist(I∗(ξ),ΓS),∀I∈W∗ Q,∀ξ∈[1,ξ 0]. (4.7) (3) For ξ>1, and recalling Definition 4.3, set U± Q=U± Q,ξ := I∈W∗,± Q I∗(ξ),UQ:=U+ Q∪U− Q,(4.8) and given S, a non-empty semi-coherent subregime of S, define ΩS:=Ω + S∪Ω− S,Ω± S=Ω ± S(ξ):=int  Q∈SU± Q.(4.9) Then there exists ξ0>1such 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 [25], it will be useful for us to extend the definition of the Whitney region UQto the case that Q ∈B, the “bad” collection of Lemma 4.5. Let W∗ Qbe the augmentation of W0 Qas constructed in Lemma 4.6, and set WQ:=W∗ Q,Q∈G, W0 Q,Q∈B .(4.10) For Q ∈Gwe shall henceforth simply write WQ, W± Qin place of W∗ Q, W∗,± Q. For arbitrary Q ∈D, good or bad, we may then make the following definitions. Definition 4.11. Given ξ>ξ>1, we let I∗=ξI and I∗ fat =ξIdenote dilated Whitney cubes, for allowable values of ξ, ξas in Definition 4.3. Suppose that x ∈∂Ωand Q ∈D. The closed Whitney region relative to Q, and its fattened version are, respectively, the sets UQ:= I∈WQ I∗,Ufat Q:= I∈WQ I∗ fat . Similarly, we define standard and fattened versions of the “semi-closed” (i.e., closed away from ∂Ω) truncated dyadic cone at x: ΥQ(x):= Q∈DQ,x∈QUQ,Υfat Q(x):= Q∈DQ,x∈QUfat Q S. Hofmann, O. Tapiola / Advances in Mathematics 390 (2021) 107961 21 and the “semi-closed” Carleson box relative to Q: TQ:= Q∈D,Q⊆QUQ,Tfat Q:= Q∈D,Q⊆QUfat Q. We list some further properties of UQand TQin 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 ηand Kand the bilateral corona decomposition (see [25, Section 3]). For an open set Ω ⊂Rn+1 that satisfies an interior corkscrew condition and has ndimensional UR boundary ∂Ω, we define the Whitney regions UQas above, but only include 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=Rn+1 \E, with for an n-dimensional UR set E=∂ΩE, as in Lemma 4.6. Lemma 4.12. Let Ω ⊂Rn+1 satisfy an interior corkscrew condition, with n-dimensional UR boundary ∂Ω. We have the following properties: •The region UQis a union of a uniformly bounded number of Whitney cubes Isuch that (Q) ≈(I)and dist(Q, I) ≈(Q). •The regions UQhave a bounded overlap property, i.e. we have i|UQi| ≈| iUQi| for cubes Qisuch that Qi=Qjif i =j. •If UQ∩UP=∅, then (Q) ≈(P)and dist(Q, P) (Q). •For every Y∈U Qwe have δ(Y) ≈(Q). •For every Q ∈D, we have |UQ| ≈(Q)n+1 ≈(Q) ·σ(Q). •If diam(∂Ω) ≈diam(Ω), then Ω =Q∈DTQ. •If diam(∂Ω) <∞and diam(Ω) =∞, then there exist Rdiam(∂Ω) and a ball B(x, R)for some x ∈∂Ωsuch that ∂Ω ⊂B(x, R)and B(x, R) ∩Ω ⊂Q∈DTQ. •If Q ∈G, then UQhas at least one connected component, and at most two, corresponding to U± Qin Lemma 4.6. •If Q ∈B, then UQhas 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 ⊂Rn+1 be as in Lemma 4.12, and write D=B∪Gas in Lemmas 4.5 and 4.6. Let Q0∈Dbe a fixed cube, denote MQ0:={Q∈B:Q⊆Q0}∪{Q(S): Q(S)⊆Q0}S∈G and set 22 S. Hofmann, O. Tapiola / Advances in Mathematics 390 (2021) 107961 GQ0(x):= Q∈MQ0 1Q(x) for every x ∈∂Ω(thus GQ0vanishes outside of Q0). Then GQ0(x) <∞for almost every x ∈Q0. In particular, for almost every x ∈Q0, there exists a stopping time regime Sx such that if x ∈Qand (Q) ≤(Q(Sx)), then Q ∈S x. For each Q ∈S x, the interior of the cone ΥQ(x)splits into at most two chord-arc domains, as does the sawtooth region ΩSx. Proof. Since the collection MQ0satisfies a Carleson packing condition by Lemma 4.5 and Q ⊂Q0for every Q ∈MQ0, we have ˆ Q0 GQ0(x)dσ(x)=  Q∈MQ0 σ(Q)σ(Q0). In particular, GQ0(x) <∞for almost every x ∈Q0. Thus, for almost every x ∈Q0there exist Cx>0such that if x ∈Qand (Q) <C x, then Q /∈MQ0. In particular, there exists a stopping time regime Sxgiven by Lemma 4.5 such that if x ∈Qand (Q) <C x, then Q ∈S x⊂G. Thus, by Lemma 4.12, the corresponding Whitney region UQsplits into at most two connected components. The final property follows now from Lemma 4.6. For every x ∈∂Ωthat satisfies the condition in Lemma 4.13, we denote the components of ΥQ(x)by Υ± Q(x), whose interiors, denoted by  Υ± Q(x), are subdomains of Ω± Sx (see (4.9)), respectively. Since Ωsatisfies the corkscrew condition, at least one of Ω± Sxis 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 ⊂Rn+1 be an open set satisfying an interior corkscrew condition and let ∂Ωbe UR. Suppose that Φ: Ω →Ris a smooth function such that μ =|∇Φ(Y)| dY is a Carleson measure, and |∇Φ(X)| 1 δ(X)for every X∈Ω. Then Φhas one-sided non-tangential boundary traces in the following sense: for σ-a.e. x ∈∂Ω, the limits ϕ+(x):= lim Y∈ Υ+ Q(Sx)(x),Y →x Φ(Y)and ϕ−(x):= lim Y∈ Υ− Q(Sx)(x),Y →x Φ(Y) exist and satisfy ϕ±L∞(∂Ω) ≤ΦL∞(Ω), provided that Ω± Sx⊂Ω. Remark 4.15. As noted above, necessarily Ω± Sx⊂Ωfor at least one choice of +or −, and possibly both. Thus, Φhas at least a 1-sided non-tangential trace a.e. on ∂Ω. In the case that both components of ΩSxare contained in Ω, the traces ϕ+and ϕ−may not coincide. Indeed, if Ω =Rn+1 +∪Rn+1 −and Φ =1 Rn+1 +−1Rn+1 −, then ϕ+(x) =1and ϕ−(x) =−1for every x ∈∂Ω(when we have chosen the directions +and −in the obvious way). S. Hofmann, O. Tapiola / Advances in Mathematics 390 (2021) 107961 23 Proof of Lemma 4.14.Fix a cube Q0∈D, and let x ∈∂Ωbe a point satisfying the condition GQ0(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⊂Ωmay be handled by the same argument. Let {Xk}kbe an arbitrary sequence of points in  Υ+ Q(Sx)(x)such that Xk→x. It suffices to show that {Φ(Xk)}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 Υ0:=Υ +,fat Q(Sx). Thus, Υ+ Q(Sx)⊂Υ0, and the interior of Υ0is an NTA domain. Let k, m ∈N, m ≥k, and let 0 <ε ξ−ξ. Since Xk, Xm∈Υ+ Q(Sx), there exists a chain of balls {Bi}N i=1, Bi:=B(Yi, ri), inside the interior of Υ0, with the following properties: (i) Y1=Xk, YN=Xm, (ii) r1≤εδ(Xk), rN≤εδ(Xm), (iii) Bi∩Bi+1 =∅for every i ≥1, (iv) ri≈δ(Yi) ≈dist(Bi, ∂Ω), (v) 1/4 ≤ri/ri+1 ≤4, (vi) for each i ≥1, Bi∪Bi+1 ⊂Ci⊂Υ0, where Ciis a cylinder with height hiand radius ρisatisfying 1≤hi/ri≤8,1≤ρi/ri≤8, and such that dist(Ci, ∂Ω) ≈diam(Ci) ≈ri, (vii) the balls {Bi}iand the cylinders {Ci}ihave bounded overlaps. Here, the implicit constants depend on the NTA properties of Υ0, and possibly on ε. We now have |Φ(Xk)−Φ(Xm)|≤¨ \ B1 |Φ(X)−Φ(Xk)|dX + N−1  i=1 ¨ \ Bi Φ(X)dX −¨ \ Bi+1 Φ(X)dX +¨ \ BN |Φ(X)−Φ(Xm)|dX :=I1+I2+I3. By the mean value theorem and the pointwise gradient bound, we know that Φis locally Lipschitz. Thus, 24 S. Hofmann, O. Tapiola / Advances in Mathematics 390 (2021) 107961 I1≤C¨ \ B1 |X−Xk| δ(Xk)dX ≤C¨ \ B1 εδ(Xk) δ(Xk)dX =Cε and similarly I3ε. As for I2, by (v) and (vi) above, and Poincaré’s inequality, we have ¨ \ Bi Φ(X)dX −¨ \ Bi+1 Φ(X)dX =¨ \ Bi Φ(X)dX −ΦCi+ΦCi−¨ \ Bi+1 Φ(X)dX ≤2¨ \ Ci Φ(X)−¨ \ Ci Φ(Y)dY dX ri |Ci|¨ Ci |∇Φ(X)|dX 1 δ(Yi)n¨ Ci |∇Φ(X)|dX ¨ Ci |∇Φ(X)|δ(X)−ndX. By construction, we may choose Q ∈S x, with (Q) ≈max(δ(Xk), δ(Xm)), such that Xk, Xm∈Υ+ Q(x)and Ci⊂Υ+,fat Q(x)for each i =1, 2, ..., N. Then, by the bounded overlap property of the cylinders {Ci}i, and the structure of the dyadic cones, we have I2 N  i=1 ¨ Ci |∇Φ(X)|δ(X)−ndX ¨ Υ+,fat Q(x) |∇Φ(X)|δ(X)−ndX ≤ x∈Q∈DQ ¨ U+,fat Q |∇Φ(X)|δ(X)−ndX  x∈Q∈DQ ¨ U+,fat Q |∇Φ(X)|(Q)−ndX  Q∈DQ 1Q(x) σ(Q)¨ U+,fat Q |∇Φ(X)|dX. We notice that ˆ Q Q∈DQ 1Q(y) σ(Q)¨ U+,fat Q |∇Φ(X)|dX dσ(y)=  Q∈DQ ¨ U+,fat Q |∇Φ(X)|dX S. Hofmann, O. Tapiola / Advances in Mathematics 390 (2021) 107961 31 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 ⊂Rn+1 is an open set with n-UR boundary ∂Ω. Suppose that fis a Borel measurable function on ∂Ω, with fL∞(∂Ω,σ)<∞. We will now construct the extension Φin Theorem 1.1. Since f∈L∞(∂Ω, σ), there is a set Z∈∂Ω, with σ(Z) =0, such that fsup(∂Ω\Z)=fL∞(∂Ω,σ). Since σis a Borel regular measure, there is a Borel set Z0⊃Z, with σ(Z0) =0. Set f0(x):=f(x),ifx∈∂Ω\Z0 0,ifx∈Z0. Note that f0=fat σ-a.e. point on ∂Ω. Moreover, f0is an everywhere bounded, Borel measurable function on ∂Ω, so by [19, Theorem 3.9.1], we know that u0:Ω →R, defined by u0(X):=ˆ ∂Ω f0(y)dωX(y), is a harmonic function in Ω satisfying u0sup(Ω) ≤f0sup(∂Ω) =fL∞(∂Ω,σ), where ωXis the harmonic measure on ∂Ωwith pole at X. Thus, by Theorem 3.1, Lemma 3.2 and (5.5), there exists a smooth 1 2-approximator of u0, i.e. a function Φ0∈ C∞(Ω) such that u0−Φ0L∞(Ω) ≤1 2u0L∞(Ω) and sup x∈∂Ω,r>0 1 rn¨ B(x,r)∩Ω |∇Φ0(Y)|dY ≤C0u0L∞(Ω), where C0depends only on dimension and the ADR and UR constants for ∂Ω. By Lemma 4.14 and Remark 4.16, Φ0has a non-tangential trace (in at least a 1-sided sense), defined σ-a.e. on ∂Ω, that we denote by ϕ0. Furthermore, by Lemma 5.6, the non-tangential trace Tu0(x) exists, with Tu0(x)=f0(x)=f(x),for σ-a.e. x∈∂Ω.(6.1) Let Z1⊂∂Ωdenote the set where either ϕ0does not exist, or where (6.1) fails, hence σ(Z1) =0. Since σis a Borel regular measure, we may assume without loss of generality that Z1is a Borel set. We now define 32 S. Hofmann, O. Tapiola / Advances in Mathematics 390 (2021) 107961 f1(x):=f0(x)−ϕ0(x),ifx∈∂Ω\Z1 0,ifx∈Z1. Then f1is an everywhere bounded Borel measurable function on ∂Ω, so there is a harmonic function u1(X):=ˆ ∂Ω f1(y)dωX(y),X∈Ω, satisfying u1L∞(Ω) ≤f1sup(∂Ω) ≤u0−Φ0L∞(Ω) ≤1 2u0L∞(Ω) ≤1 2fL∞(∂Ω). Again using Theorem 3.1 and Lemma 3.2, we may construct a smooth 1 2-approximator of u1, i.e. a function Φ1∈C∞(Ω) such that u1−Φ1L∞(Ω) ≤1 2u1L∞(Ω) ≤1 4u0L∞(Ω),and sup x∈∂Ω,r>0 1 rn¨ B(x,r)∩Ω |∇Φ1(Y)|dY ≤C0u1L∞(Ω) ≤1 2C0u0L∞(Ω) , with C0as above. By Lemma 4.14 and Remark 4.16, Φ1has a non-tangential trace (in at least a 1-sided sense), defined σ-a.e. on ∂Ω, that we denote by ϕ1. Moreover, by Lemma 5.6, u1has a non-tangential trace Tu1such that Tu1(x)=f1(x)=f0(x)−ϕ0(x),σ-a.e. x∈∂Ω.(6.2) Let Z2⊂∂Ωbe the set of σ-measure 0such that either (6.2) fails, or ϕ1does not exist. Again, without loss of generality, we may assume that Z2is a Borel set. We set f2(x):=f1(x)−ϕ1(x)=f0(x)−ϕ0(x)−ϕ1(x),ifx∈∂Ω\Z2 0,ifx∈Z2. We let u2be the harmonic extension of f2, and iterate, to obtain for each k∈N0, a sequence of Borel sets Zk⊂∂Ωof σ-measure 0, harmonic functions uk, their 1 2approximators Φk, the non-tangential boundary traces ϕkof the approximators, and the non-tangential boundary traces fk+1 of the function uk−Φk. These satisfy (i) fk+1 =f0(x) −k i=0 ϕi(x), x ∈∂Ω \Zk+1, (ii) fk+1sup(∂Ω) ≤uk−ΦkL∞(Ω) ≤2−k−1u0L∞(Ω) ≤2−k−1f0sup(∂Ω), (iii) ukL∞(Ω) ≤fksup(∂Ω) ≤2−ku0L∞(Ω) (iv) supx∈∂Ω,r>01 rn˜B(x,r)∩Ω|∇Φk(Y)| dY ≤C0ukL∞(Ω) ≤2−kC0u0L∞(Ω). (v) ΦkL∞(Ω) 2−ku0L∞(Ω) (by (ii), (iii) and the triangle inequality). S. Hofmann, O. Tapiola / Advances in Mathematics 390 (2021) 107961 33 By (v), we may define the uniformly convergent series Φ(X):=∞  k=0 Φk(X),X∈Ω.(6.3) By construction, the function Φhas a non-tangential boundary trace ϕ(in at least a 1-sided sense; we recall that the 1-sided approach may be taken to be the same for all Φk: see Remark 4.16), defined σ-a.e. on ∂Ω, ϕ(x)= ∞  k=0 ϕk(x). Since limk→∞ fksup(∂Ω) ≤limk→∞ 2−kf0sup(∂Ω) =0, we have limk→∞ fk(x) =0for every x ∈∂Ω. In particular, by (i) above we have 0 = lim k→∞fk+1(x) = lim k→∞f0(x)− k  i=0 ϕi(x)=f0(x)−ϕ(x),σ-a.e. x∈∂Ω (that is, for x ∈∂Ω \(∪kZk)). Thus, ϕ(x) =f(x)for σ-a.e. x ∈∂Ω, since f0=fat σ-a.e. point on ∂Ω. Also, for x ∈∂Ωand r>0, and for every −→ Ψ∈C1 0(B(x, r) ∩Ω) satisfying −→ ΨL∞≤1, using (6.3)and then (iv), we have 1 rn¨ B(x,r)∩Ω Φ(Y)div −→ Ψ(Y)dY =∞  k=0 1 rn¨ B(x,r)∩Ω Φk(Y)div −→ Ψ(Y)dY ≤1 rn¨ B(x,r)∩Ω |∇Φk(Y)|dY ≤∞  k=0 2−kC0u0L∞(Ω) =2C0u0L∞(Ω). Thus, the measure μ :=|∇Φ(Y)| dY is a Carleson measure. By Lemmas 3.6, 3.8, 3.13 and 3.14, we may further assume that Φ ∈C∞(Ω), and that |∇Φ(X)| u0L∞(Ω)δ(X)−1. Since u0L∞(Ω) ≤fL∞(∂Ω), this completes the proof of Theorem 1.1. Remark 6.4. Note that the preceeding argument involved the construction of a bounded harmonic extension u, corresponding to given Borel measurable data f∈L∞(∂Ω, dσ), such that the non-tangential trace Tusatisfies Tu(x) =f(x)for σ-a.e. x ∈∂Ω. It is perhaps worthwhile to observe that, in the absence of absolute continuity of harmonic measure with respect to σ, this extension need not be unique. Indeed, suppose that fsup(∂Ω\Z)=fL∞(∂Ω,σ)=1, for a Borel set Z⊂∂Ωwith σ(Z) =0. Set 34 S. Hofmann, O. Tapiola / Advances in Mathematics 390 (2021) 107961 g0(x):=f(x),ifx∈∂Ω\Z 0,ifx∈Z, g1(x):=f(x),ifx∈∂Ω\Z 1,ifx∈Z, and define vi(Y):=vgi(Y):=ˆ ∂Ω gidωY,Y∈Ω,i=0,1. Then viL∞(Ω) ≤1for i =0, 1, and by Lemma 5.6, the traces Tv0and Tv1exist σ-a.e. on ∂Ω, and satisfy Tv0=g0=f=g1=Tv1,σ-a.e. on ∂Ω. On the other hand, v1(Y)=v0(Y)+ωY(Z), so if harmonic measure has positive mass on Z, then v1=v0. 7. 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 [25, 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 [25], we start by presenting a simplified construction of the boxes and proving that the boundaries of the boxes inside Ωare upper n-ADR. We note that the original proof for the upper n-ADR property of the boundaries of Carleson boxes in [25, Appendix] does not apply “off-the-shelf” in our situation because we do not use dilated (hence overlapping) Whitney cubes (as is done in [25, Appendix]). However, our approach makes the proof quite simple. In this section, Ω ⊂Rn+1 is an open set, satisfying the corkscrew condition, with d-ADR boundary ∂Ωfor some d ∈(0, n], and Dis a dyadic system on ∂Ω. Recall the Whitney decomposition and the definition of the collections WQ=WQ(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∈Wis assumed to be of the form I=Π n+1 k=1(ak, ak+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: S. Hofmann, O. Tapiola / Advances in Mathematics 390 (2021) 107961 35 Fig. 1. A rough idea of the structure of the Carleson box TQ(left) and the Carleson tent τQ(right) on top of a same cube Qin the simplest case where Ω =R2 +. Lemma 7.2. We can choose the parameters ηand Kdepending only on the corkscrew constants, so that WQ=∅for every Q ∈D. The proof is a straightforward generalization of [25, Remark 3.3] and [22, Lemma 5.3]. We omit the details. Let us remark that in the codimension 1 case, if Ω =Rn+1 \E, with En-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 =Rn+1 \Ehas only one connected component, which necessarily satisfies the corkscrew condition. Definition 7.3. Suppose that x ∈∂Ωand Q ∈D. The “half-open” Whitney region relative to Qis the set UQ:= I∈WQ I, the dyadic cone at xis the set Γ(x):= Q∈D:x∈Q UQ the Carleson box relative to Qis the set (see Fig. 1) TQ:= Q∈D,Q⊆Q UQ and the Carleson tent relative to Qis the set (see Fig. 1) τQ:=Ω\ y∈∂Ω\Q Γ(y) Remark 7.4. We note that every I∈Wwith (I) diam(∂Ω) belongs to the collection WQI, where as above (QI) =(I) ≈dist(I, QI), and QIis chosen to minimize 36 S. Hofmann, O. Tapiola / Advances in Mathematics 390 (2021) 107961 dist(I, QI). Moreover, for ηchosen small enough and Klarge enough depending only on the properties of the Whitney decomposition, every J∈Wwhose closure touches the closure of I, also belongs to WQI. Consequently, for such ηand K, we have: •if diam(∂Ω) <∞and diam(Ω) =∞, then Q∈DTQ⊃B(x, R) ∩Ωfor some point x ∈∂Ωand R≈diam(∂Ω), •if diam(∂Ω) ≈diam(Ω), then Q∈DTQ⊃Ω. Remark 7.5. Given m ∈(1, ∞), one may choose ηsmall enough and Klarge enough, depending on m, so that the dyadic cone Γ(x)contains (at least locally) a cone of the type  Γm(x) ={Y∈Ω: dist(x, Y) <mδ(Y)}; i.e.,  Γm(x) ∩B(x, R) ⊂Γ(x)for R≈diam(∂Ω). We omit the routine proof of this fact. We now fix a suitably large aperture constant mthat 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 ηand Kdepending only on the corkscrew constants, the Whitney cube constants, and the fixed aperture parameter m, in such a way that the collections WQare nonempty, the Carleson boxes TQhave good covering properties and the dyadic cones contain “regular” cones. The sets UQ, TQand Γ(x)then satisfy the same properties (with possibly different implicit constants) as UQ, TQand ΥQ(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 TQin Ωare upper n-ADR. The boundaries of the boxes constructed in [25]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 <∞  Q∈DQ dist(Q,Qc)≤κ(Q) I∈WQHn(∂I)≤Cκ(Q)n.(7.7) Proof. Note that the number of Whitney cubes in WQis uniformly bounded for each Q, and that for I∈W Qwe have Hn(∂I) ≈(Q)n, by the definition of WQ; consequently  I∈WQHn(∂I)(Q)n. Organizing the subcubes of Qby dyadic generation DQ=∪∞ k=0Dk Q, where Dk Q:={Q⊂Q:(Q)=2 −k(Q)},0≤k≤∞, we obtain by the thin boundary property (Theorem 2.16 (v)) that S. Hofmann, O. Tapiola / Advances in Mathematics 390 (2021) 107961 37  Q∈Dk Q dist(Q,Qc)(Q) σ(Q)2−kγσ(Q).(7.8) Combining these observations, we obtain in the codimension 1 case d =nthat ∞  k=0  Q∈Dk Q dist(Q,Qc)(Q) I∈WQHn(∂I)∞  k=0 2−kγσ(Q)σ(Q), or in general that ∞  k=0  Q∈Dk Q dist(Q,Qc)(Q) I∈WQHn(∂I)∞  k=0  Q∈Dk Q dist(Q,Qc)(Q) (Q)n ≤(Q)n−d∞  k=0  Q∈Dk Q dist(Q,Qc)(Q) (Q)d ≈(Q)n−d∞  k=0  Q∈Dk Q dist(Q,Qc)(Q) σ(Q) (Q)n−d∞  k=0 2−kγσ(Q)(Q)n. Lemma 7.9. For each Q, the set  ∂TQis upper n-ADR, where  ∂TQ:=∂TQ∩Ω: for every X∈ ∂TQand every R∈(0, diam(TQ)) we have Hn( ∂TQ∩B(X, R)) Rn, 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∈ ∂TQ, then by construction there exists a dyadic cube Q∈DQ and a Whitney cube I∈W Qsuch that X∈∂I. Also, if (Q) (Q)and dist(Q, Qc)  (Q)for Q∈DQ, then ∂I ∩ ∂TQ=∅for every I∈W Q. Thus, if I⊂TQ, with ∂I ∩ ∂TQ=∅, then I∈W Qfor a cube Q∈DQsuch that dist(Q, Qc) (Q), where the implicit constant depend on ηand 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 Hn( ∂TQ)≤ Q∈DQ dist(Q,Qc)(Q) I∈WQHn(∂I)(Q)n. 38 S. Hofmann, O. Tapiola / Advances in Mathematics 390 (2021) 107961 Thus, we have Hn( ∂TQ) (Q)n≈diam(Q)nfor any Q ∈D. Let us then prove the upper n-ADR property. Suppose that X∈ ∂TQand R∈(0, diam(TQ)). There are three cases: 1) Suppose that R≈diam(TQ). Then, by the consideration above, we have Hn( ∂TQ∩B(X, R)) ≤H n( ∂TQ)diam(Q)n≈diam(TQ)n≈Rn. 2) Suppose that Rδ(X). Then, by construction, B(X, R) ∩ ∂TQis contained in a union of a uniformly bounded number of boundaries of Whitney cubes Isuch that (I) >R. Since ∂I is clearly n-ADR for each I∈W, we therefore find that Hn( ∂TQ∩B(X, R)) Rn. 3) Suppose that δ(X) Rdiam(TQ). Then  ∂TQ∩B(X, R) = ∂TQ∩B(X, R) for some subcube of Q∈DQwith (Q) ≈R. Thus, by the consideration above, we have Hn( ∂TQ∩B(X, R)) = Hn( ∂TQ∩B(X, R)) ≤H n( ∂TQ)(Q)n≈Rn. This completes the proof.  8. Modified Carleson tents Fix a cube Q0∈D. For all Q ⊆Q0, we shall now construct disjoint Carleson tents tQ, that have better covering properties than τQ. We let {Q0}be “generation zero”, and then enumerate the dyadic descendants of Q0: let {Qi 1}ibe the first generation of descendants, {Qi 2}ithe second generation of descendants, and so on. Let the number of descendants of generation kbe N(k). We construct a restricted version of the Whitney collection WQ, Q ⊂Q0, by removing some of the cubes from WQi k: for each k, i ∈N, i ≤N(k), we set Wr Qi k:=WQi k\⎛ ⎝ k−1  m=0 N(m)  j=1 WQj m∪ i−1  j=1 WQj k⎞ ⎠, where of course the second union is vacuous if i =1, and both are vacuous if k=0. Note that the restricted Whitney collections {Wr Q}Q⊂Q0are pairwise disjoint, by construction. We can then define restricted Whitney regions Ur Qand modified Carleson tents tQfor cubes Q ⊆Q0(see Fig. 2): Ur Q:= I∈Wr Q I, tQ:= Q∈D,Q⊆Q Ur Q(8.1) S. Hofmann, O. Tapiola / Advances in Mathematics 390 (2021) 107961 39 Fig. 2. Two modified Carleson tents tQand tQin the simplest case where Ω =R2 +. 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 {Wr Q}Q⊂Q0are pairwise disjoint, and since we are now working with half-open (hence disjoint) Whitney cubes I, it follows that the sets {Ur Q}Q⊂Q0are also pairwise disjoint. Lemma 8.3. Suppose that Q, Q1, Q2∈DQ0. We then have: i) τQ⊂tQ. ii) If Q1∩Q2=∅, then also tQ1∩tQ2=∅. iii) If Q1⊂Q2, then also tQ1⊂tQ2. iv) TQ0=tQ0. Moreover, for Q Q0, there is a collection F(Q) ={Qi}N i=1 ⊂DQ0, of uniformly bounded cardinality Ndepending only on n, ADR, ηand K, such that (Qi) ≈η,K (Q)with (Qi) =(Qi)for all i, i, and TQ⊂itQi. 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),  y∈∂Ω\Q Γ(y)=  Q∈D\DQ UQ, and that Ur Q⊂UQfor every Q∈DQ0. Moreover, the restricted Whitney regions Ur Q are disjoint (see Remark 8.2). Consequently, τQ=Ω\ y∈∂Ω\Q Γ(y)=TQ\ Q∈D\DQ UQ⊂TQ\ Q∈DQ0\DQ Ur Q⊂tQ. Lemma 8.4. The sets ∂tQ∩Ωare upper n-ADR with the ADR constant depending only on the dimension and the ADR constant of ∂Ω. Proof. Recall that τQ⊂tQ, by Lemma 8.3 i). Thus, if I⊂tQ, with ∂I ∩∂tQ=∅, then I∈W r Qfor a cube Q∈DQsuch that dist(Q, Qc) (Q). One may then use Lemma 7.6, following the proof of Lemma 7.9 with minor adjustments. We omit the details.  40 S. Hofmann, O. Tapiola / Advances in Mathematics 390 (2021) 107961 9. Proof of Proposition 1.3 Suppose that Ω ⊂Rn+1 is an open set satisfying the corkscrew condition with d-ADR boundary for some d ∈(0, n]. Let Q0∈Dbe a fixed dyadic cube,  DQ0={Qj}j⊂DQ0 be a collection of subcubes of Q0and {αj}ja collection of coefficients such that f(x):= j αj1Qj, belongs to BMO(∂Ω), the collection  DQ0enjoys a Carleson packing condition with packing norm C DQ0=:C0(see Definition 2.23), and supj|αj| fBMO. Note that fvanishes on ∂Ω \Q0, but we assume that f∈BMO, globally on ∂Ω. We denote F0:= j αj1tQj, where tQjis the modified Carleson tent defined in (8.1). We will show that a smooth version of F0satisfies the properties in Proposition 1.3. We start by proving the following estimate that we shall need later: Lemma 9.1. Let Q, Q∈Dbe such that (Q)≈(Q)dist(Q, Q).(9.2) Then  j:Qj⊇Q αj− j:Qj⊇Q αj C0fBMO, 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 Clarge enough (depending only on the implicit constants in (9.2)) that Q ∪Q⊂B∗ Q:=B(xQ, r), with r:=C(Q). Let Δ∗ Q:=B∗ Q∩∂Ωdenote the corresponding surface ball. Since f∈BMO(∂Ω), by the ADR property we have Q |f−fΔ∗ Q|+ Q|f−fΔ∗ Q| Δ∗ Q |f−fΔ∗ Q|≤fBMO .(9.3) By the uniform bound on the coefficients and the packing condition of the collection {Qj}j, we have that S. Hofmann, O. Tapiola / Advances in Mathematics 390 (2021) 107961 47 We may then generalize Lemma 8.3, so that in particular, for each Q ∈DP1∪... ∪DPN, there is a collection F(Q) ={Qi}i⊂DQ0∪DP1∪... ∪DPN, of uniformly bounded cardinality depending only on n, ADR, ηand K, such that (Qi) ≈(Q), with (Qi) = (Qi)for all i, i, and TQ⊂it∗ Qi. Moreover, t∗ Q⊂t∗ Q , provided that Q⊂Q, and t∗ Q∩t∗ Q =∅whenever Q∩Q =∅. One may now repeat the previous argument, mutatis mutandis, noting that Lemma 9.1 still applies in the case that Qi∩Q0=∅. We omit the details.  10. Garnett’s decomposition lemma and proof of Theorem 1.2 In this last section, we present the final ingredient for the proof of Theorem 1.2: a straightforward generalization of Garnett’s decomposition lemma to the setting of ADR sets. The proof follows the original argument sketched as an exercise in Garnett [18, Section VI, Exercise 12 c](and stated without proof in [41, Lemma 1.2.1]). We include the details here for the sake of completeness. Lemma 10.1 (Garnett’s lemma). Let E⊂Rn+1 be a d-ADR set, d ≤n. Let Q0∈D, and consider f∈BMOD(E) (see Definition 2.3), which vanishes on E\Q0(provided the latter is non-empty). Then there is a collection  DQ0={Qj}j⊂DQ0and coefficients αjsuch that (1) supj|αj| fBMOD, (2) f−fQ0= f+jαj1Qj, where  f∈L∞(E, dσ)with  fL∞fBMOD, (3)  DQ0satisfies a Carleson packing condition with C DQ0 1. Remark 10.2. i) Since fBMODfBMO, the Lemma holds of course for f∈BMO(E). ii) The construction of the coefficients αjis based on the same arguments as the proof of the John–Nirenberg lemma [33]. Remark 10.3. If Q0E, then there is a cube Q1disjoint from Q0, of the same dyadic generation (i.e., such that (Q1) =(Q0)), with common dyadic ancestor Q∗, such that dist(Q0, Q1) (Q0) =(Q1) ≈(Q∗). Since fvanishes outside of Q0, we have that f≡0on Q1, hence |fQ0|=|fQ0−fQ1|fBMOD, where the last inequality is a well-known fact about dyadic BMO. Consequently, in this case we may absorb fQ0into  f, so that item (2) in Lemma 10.1 becomes (2a) f= f+jαj1Qj, where  f∈L∞(E, dσ), with  fL∞fBMOD. 48 S. Hofmann, O. Tapiola / Advances in Mathematics 390 (2021) 107961 Observe also that in this case  fvanishes on E\Q0. Proof of Lemma 10.1.We build the collection  DQ0by using a stopping time argument. Set F0={Q0}. We have |f−fQ0|Q0≤fBMOD. Let us subdivide Q0and stop when |f−fQ0|Q>2fBMOD. We let F1={Q(1) j}j be the collection of the maximal stopping cubes. By definition, |f−fQ(1) j|Q(1) j≤fBMOD,∀Q(1) j∈F 1. For each Q(1) j, we repeat the process with the modified stopping condition |f−fQ(1) j|Q>2fBMOD. We let F2={Q(2) j}jbe the collection of maximal stopping cubes. Again by definition, |f−fQ(2) j|Q(2) j≤fBMOD. We continue in this way, and denote the collection of cubes of level iby Fi. We now set  DQ0:=iFi, and define α(i) j:=f−fP(i−1) k(j)Q(i) j=fQ(i) j−fP(i−1) k(j) , where for i ≥1, P(i−1) k(j)is the unique cube in Fi−1such that Q(i) j⊂P(i−1) k(j). We prove the properties (1) – (3) in order. (1) Property (1) follows easily from the ADR property and the stopping criterion: |α(i) j|= Q(i) j f−fP(i−1) k(j) dσ≤ Q(i) j f−fP(i−1) k(j)dσ   Q(i) j f−fP(i−1) k(j)dσ fBMOD, where  Q(i) jis the dyadic parent of Q(i) j. (2) Observe that f−fQ0=−fQ0in E\Q0, if the latter is non-empty, and in this case, by Remark 10.3 we may simply set  f=−fQ0on E\Q0. It is therefore enough to prove the decomposition (2) on Q0. S. Hofmann, O. Tapiola / Advances in Mathematics 390 (2021) 107961 49 For x ∈Q0, we define a counting function Nf(x):=#i≥1:∃Q(i) j∈F iwith x∈Q(i) j. If Nf(x) <∞, we set Nx:=Nf(x), and note that in this case there is a cube Qmin(x) ∈ FNxsuch that x ∈Qmin(x), and x /∈Q(i) jfor all i >N xand every j. Also, for every i ≤Nx, there now exists a cube Q(i) j(i,x)∈F isuch that x ∈Q(i) j(i,x). Since the cubes in each Fiare disjoint, by the definition of the cubes P(i−1) k(j), we have α(i) j(i,x)=fQ(i) j(i,x)−fP(i−1) k(j(i,x)) =fQ(i) j(i,x)−fQ(i−1) j(i−1,x) . In particular, the sum Nx i=1 α(i) j(i,x)is telescoping and we get  i,j α(i) j1Q(i) j(x)= Nx  i=1 α(i) j(i,x)=−fQ0+fQmin(x). On the other hand, if Nf(x) =∞, then the analogous telescoping sum becomes  i,j α(i) j1Q(i) j(x)= ∞  i=1 α(i) j(i,x)=−fQ0+f(x), by Lebesgue’s differentiation theorem, where the latter identity is valid for σ-a.e. xsuch that Nf(x)is infinite. Setting  f(x):=f(x)−fQmin(x),if Nf(x)<∞ 0,if Nf(x)=∞, we obtain the claimed decomposition in (2). It remains to check that with this definition, we have  fL∞(E,dσ)fBMOD. To this end, observe that in order to have Nf(x) <∞, we must have that for every dyadic cube Qwith x ∈Q Qmin(x), f−fQmin(x)Q≤2fBMOD, otherwise, there would have been another stopping cube containing x, and strictly contained in Qmin(x), which contradicts the definition of Qmin(x). By Lebesgue’s differentiation theorem, we therefore find that | f(x)| ≤2fBMODfor σ-a.e. xsuch that Nf(x) <∞, so that (2) holds. (3) By a standard limiting argument, we may assume that the collection  DQ0is finite. We first notice that by the stopping conditions we have 50 S. Hofmann, O. Tapiola / Advances in Mathematics 390 (2021) 107961 σ(Q(i) j)≤1 2fBMODˆ Q(i) j |f−fP(i−1) k(j)|dσ. (10.4) Let Q ⊆Q0be fixed. We set I:= R∈ DQ0,R⊆Q σ(R)=I1+I2, where I1is the sum over those Q(i) jsuch that P(i−1) k(j)⊂Qand I2is the sum over the rest of the relevant cubes. The cubes in the sum I2are disjoint and thus, I2≤σ(Q). Let i(Q) be the smallest integer such that Fi(Q)contains at least one cube in the sum I1; thus, I2is the sum over the cubes in Fi(Q)−1that are contained in Q. With this notation, we may write I= i≥i(Q)−1 R∈Fi,R⊂Q σ(R)=  i≥i(Q) R∈Fi,R⊂Q σ(R)+I2=I1+I2. We have I1= i≥i(Q) j:Q(i) j∈Fi,Q(i) j⊂Q σ(Q(i) j) (10.4) ≤1 2fBMOD i≥i(Q) j:Q(i) j∈Fi,Q(i) j⊂Q ˆ Q(i) j |f−fP(i−1) k(j)|dσ (A) =1 2fBMOD i≥i(Q) j:Q(i−1) j∈Fi−1,Q(i−1) j⊂Q l:Q(i) l∈Fi,Q(i) l⊂Q(i−1) j ˆ Q(i) l |f−fQ(i−1) j|dσ (B) ≤1 2fBMOD i≥i(Q) j:Q(i−1) j∈Fi−1,Q(i−1) j⊂Q ˆ Q(i−1) j |f−fQ(i−1) j|dσ =1 2fBMOD i≥i(Q) j:Q(i−1) j∈Fi−1,Q(i−1) j⊂Q σ(Q(i−1) j)|f−fQ(i−1) j|Q(i−1) j (C) ≤1 2 i≥i(Q) j:Q(i−1) j∈Fi−1,Q(i−1) j⊂Q σ(Q(i−1) j) ≤1 2I, where we used in (A) the observation that with this notation P(i−1) k(l)=Q(i−1) j, in (B) the fact that the cubes Q(i) l(j)∈F iare disjoint, and in (C) the definition of the BMOD norm. In particular, S. Hofmann, O. Tapiola / Advances in Mathematics 390 (2021) 107961 51 I=I1+I2≤1 2I+σ(Q) and thus I≤2σ(Q). This completes the proof.  Theorem 1.2 follows now easily from the other results we have proven: Proof of Theorem 1.2.Suppose that fis a compactly supported function in BMO(∂Ω). Then, by Theorem 2.16, there is a choice of dyadic system Dsuch that there exists a cube Q0∈Dwith supp f⊂Q0. By Lemma 10.1, there exists now a decomposition f= f+f0, where (1)  fis bounded σ-a.e., and  fL∞(∂Ω) fBMO(∂Ω), and (2) f0(x) =Q∈ DQ0αQ1Q(x)for a collection  DQ0⊂DQ0and coefficients αQsuch that •C DQ0 1, and •supQ∈ DQ0|αQ| fBMO(∂Ω). By Theorem 1.1, we know that there exists a function Φ ∈C∞(Ω) such that Φconverges to  fnon-tangentially almost everywhere, the measure μ1:=|∇Φ(Y)| dY is a Carleson measure and Cμ1 fL∞(∂Ω) fBMO(∂Ω). By the decomposition f= f+f0, we know that f0is 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 Fconverges to f0non-tangentially almost everywhere, the measure μ2:= |∇F(Y)| dY is a Carleson measure and Cμ2C DQ0f0BMO(∂Ω)  fBMO(∂Ω) +fBMO(∂Ω)  fL∞(∂Ω) +fBMO(∂Ω) fBMO(∂Ω). Thus, we can set V:=Φ +F. References [1] T.C. Anderson, T. Hytönen, O. Tapiola, Weak A∞weights and weak reverse Hölder property in a space of homogeneous type, J. Geom. Anal. 27 (1) (2017) 95–119. [2] J. Azzam, Semi-uniform domains and the A∞property for harmonic measure, Int. Math. Res. Not. IMRN (9) (2021) 6717–6771. [3] J. Azzam, J. Garnett, M. Mourgoglou, X. Tolsa, Uniform rectifiability, elliptic measure, square functions, and ε-approximability via an acf monotonicity formula, preprint, arXiv :1612 .02650, 2016. [4] J. Azzam, S. Hofmann, J.M. Martell, M. Mourgoglou, X. Tolsa, Harmonic measure and quantitative connectivity: geometric characterization of the Lp-solvability of the Dirichlet problem, Invent. Math. 222 (3) (2020) 881–993. [5] C.J. Bishop, P.W. Jones, Harmonic measure and arclength, Ann. Math. (2) 132 (3) (1990) 511–547. [6] S. Bortz, S. Hofmann, Quantitative Fatou theorems and uniform rectifiability, Potential Anal. 53 (1) (2020) 329–355. [7] S. Bortz, O. Tapiola, ε-approximability of harmonic functions in Lpimplies uniform rectifiability, Proc. Am. Math. Soc. 147 (5) (2019) 2107–2121. 52 S. Hofmann, O. Tapiola / Advances in Mathematics 390 (2021) 107961 [8] L. Carleson, Interpolations by bounded analytic functions and the corona problem, Ann. Math. (2) 76 (1962) 547–559. [9] M. Christ, A T(b)theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (2) (1990) 601–628. [10] D. Cruz-Uribe, Two weight inequalities for fractional integral operators and commutators, in: Advanced Courses of Mathematical Analysis VI, World Sci. Publ., Hackensack, NJ, 2017, pp. 25–85. [11] G. David, D. Jerison, Lipschitz approximation to hypersurfaces, harmonic measure, and singular integrals, Indiana Univ. Math. J. 39 (3) (1990) 831–845. [12] G. David, S. Semmes, Singular integrals and rectifiable sets in Rn: beyond Lipschitz graphs, Astérisque (193) (1991) 152. [13] G. David, S. Semmes, Analysis of and on Uniformly Rectifiable Sets, Mathematical Surveys and Monographs, vol. 38, American Mathematical Society, Providence, RI, 1993. [14] M. Dindos, C. Kenig, J. Pipher, BMO solvability and the A∞condition for elliptic operators, J. Geom. Anal. 21 (1) (2011) 78–95. [15] L.C. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. [16] C. Fefferman, E.M. Stein, Hpspaces of several variables, Acta Math. 129 (3–4) (1972) 137–193. [17] J. Garnett, M. Mourgoglou, X. Tolsa, Uniform rectifiability from Carleson measure estimates and ε-approximability of bounded harmonic functions, Duke Math. J. 167 (8) (2018) 1473–1524. [18] J.B. Garnett, Bounded Analytic Functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. [19] L.L. Helms, Potential Theory, second edition, Universitext, Springer, London, 2014. [20] S. Hofmann, Quantitative absolute continuity of harmonic measure and the Dirichlet problem: a survey of recent progress, Acta Math. Sin. Engl. Ser. 35 (6) (2019) 1011–1026. [21] S. Hofmann, P. Le, BMO solvability and absolute continuity of harmonic measure, J. Geom. Anal. 28 (4) (2018) 3278–3299. [22] S. Hofmann, J.M. Martell, Uniform rectifiability and harmonic measure I: uniform rectifiability implies Poisson kernels in Lp, Ann. Sci. Éc. Norm. Supér. (4) 47 (3) (2014) 577–654. [23] S. Hofmann, O. Tapiola, Uniform rectifiability and ε-approximability of harmonic functions in Lp, Ann. Inst. Fourier (Grenoble) 70 (4) (2020) 1595–1638. [24] S. Hofmann, P. Le, J.M. Martell, K. Nyström, The weak-A∞property of harmonic and p-harmonic measures implies uniform rectifiability, Anal. PDE 10 (3) (2017) 513–558. [25] S. Hofmann, J.M. Martell, S. Mayboroda, Uniform rectifiability, Carleson measure estimates, and approximation of harmonic functions, Duke Math. J. 165 (12) (2016) 2331–2389. [26] S. Hofmann, J.M. Martell, S. Mayboroda, Transference of scale-invariant estimates from Lipschitz to non-tangentially accessible to uniformly rectifiable domain, preprint, arXiv :1904 .13116, 2019. [27] S. Hofmann, C. Kenig, S. Mayboroda, J. Pipher, Square function/non-tangential maximal function estimates and the Dirichlet problem for non-symmetric elliptic operators, J. Am. Math. Soc. 28 (2) (2015) 483–529. [28] S. Hofmann, D. Mitrea, M. Mitrea, A.J. Morris, Square function estimates in spaces of homogeneous type and on uniformly rectifiable Euclidean sets, Electron. Res. Announc. Math. Sci. 21 (2014) 8–18. [29] T. Hytönen, A. Kairema, Systems of dyadic cubes in a doubling metric space, Colloq. Math. 126 (1) (2012) 1–33. [30] T. Hytönen, A. Rosén, Bounded variation approximation of Lpdyadic martingales and solutions to elliptic equations, J. Eur. Math. Soc. 20 (8) (2018) 1819–1850. [31] T. Hytönen, O. Tapiola, Almost Lipschitz-continuous wavelets in metric spaces via a new randomization of dyadic cubes, J. Approx. Theory 185 (2014) 12–30. [32] D.S. Jerison, C.E. Kenig, Boundary behavior of harmonic functions in nontangentially accessible domains, Adv. Math. 46 (1) (1982) 80–147. [33] F. John, L. Nirenberg, On functions of bounded mean oscillation, Commun. Pure Appl. Math. 14 (1961) 415–426. [34] C. Kenig, H. Koch, J. Pipher, T. Toro, A new approach to absolute continuity of elliptic measure, with applications to non-symmetric equations, Adv. Math. 153 (2) (2000) 231–298. [35] C.E. Kenig, Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, CBMS Regional Conference Series in Mathematics, vol. 83, 1994, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI. [36] F. Riesz, M. Riesz, Über die randwerte einer analytischen funktion, in: Comptes Rendus du Quatrième Congrès des Mathématiciens Scandinaves, Stockholm, 1916, Almqvists and Wilksels, Uppsala, 1920, pp. 27–44. S. Hofmann, O. Tapiola / Advances in Mathematics 390 (2021) 107961 53 [37] E. Sawyer, R.L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Am. J. Math. 114 (4) (1992) 813–874. [38] S.W. Semmes, A criterion for the boundedness of singular integrals on hypersurfaces, Trans. Am. Math. Soc. 311 (2) (1989) 501–513. [39] E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, vol. 30, Princeton University Press, Princeton, N.J., 1970. [40] O. Tapiola, Adjacent dyadic systems and the Lp-boundedness of shift operators in metric spaces revisited, Colloq. Math. 145 (1) (2016) 121–135. [41] N.Th. Varopoulos, BMO functions and the ∂-equation, Pac. J. Math. 71 (1) (1977) 221–273. [42] N.Th. Varopoulos, A remark on functions of bounded mean oscillation and bounded harmonic functions. Addendum to: “BMO functions and the ∂-equation” (Pacific J. Math. 71 (1977), no. 1, 221–273), Pac. J. Math. 74 (1) (1978) 257–259. [43] Z. Zhao, BMO solvability and A∞condition of the elliptic measures in uniform domains, J. Geom. Anal. 28 (2) (2018) 866–908.