Potential Analysis(2020) 53:1025–1041 https://doi.org/10.1007/s11118-019-09796-6
Dimension Drop for Harmonic Measure on Ahlfors Regular Boundaries
Jonas Azzam1
Received: 17 December 2018 / Accepted: 4 August 2019 /Published online: 19 August 2019 © The Author(s) 2019
Abstract We show that given a domain ⊆ Rd+1 with uniformly non-flat Ahlfors s-regular boundary with s ≥ d, the dimension of its harmonic measure is strictly less than s.
Keywords Harmonic measure · Dimension · Ahlfors regular sets
Mathematics Subject Classification (2010) 31A15 · 28A75 · 28A78 · 31B05 · 35J25
1 Introduction
The purpose of this note is to study the dimension of harmonic measure ω for a connected d+1 domain ⊆ R . Naturally, dim ω ≤ dim ∂, but the inequality can be strict. For example, it is a classical result of Jones and Wolff [22]thatif ⊆ C, then the dimension is always at most 1, even if dim ∂ > 1 (which improves on an earlier result of Makarov for simply connected planar domains [25]). In higher dimensions, the analogous property is no longer true: there are domains called Wolff snowflakes in Rd+1 whose harmonic measure can be strictly larger or strictly less than d;thed = 2 case is due to Wolff [34], and the general case is Lewis, Verchota and Vogel in [24] (note that even though the dimension can be above d, a result of Bourgain says that the dimension harmonic measure for any domain in Rd+1 can’t get too close to d + 1[14], and it is an open problem to determine what the supremal dimension can be). While these are all very non-trivial results, these Wolff snowflakes actually have some nice geometry. In particular, they are two-sided uniform domains. We say a domain is C-uniform if for all x,y ∈ there is a curve γ ⊆ so that H 1(γ ) ≤ C|x − y| and − dist(z, c) ≥ C 1 min{(x, z), (y, z)} where (a, b) denotes the length of the subarc of γ between a and b. A domain is two-sided c uniform if both and are C-uniform domains for some C.
Jonas Azzam [email protected]
1 School of Mathematics, University of Edinburgh, JCMB, Kings Buildings, Mayfield Road, Edinburgh, EH9 3JZ, Scotland 1026 J. Azzam
Since two-sided uniform domains have boundaries with dimension at least d and a Wolff snowflake can have dimension less than d, this example also shows that a dimension drop for harmonic measure can occur for a domain that is quite nice in terms of its connectivity and boundary properties. Hence, it is an interesting problem to identify some general criteria for when a whole class of domains satisfy dim ω < dim ∂. A dimension drop for harmonic measure occurs for some domains whose boundaries have some self-similar structure. This phenomenon was first observed by Carleson [15]for complements of planar Cantor sets whose boundaries have dimension at least 1 (rather, he showed for a particular class of Cantor sets C,dimω < 1). Later, Jones and Wolff showed the same result but for uniformly perfect sets satisfying a certain uniform disconnectedness property (see [22]or[18, Section X.I.2]). Makarov and Volberg showed dim ω < dim ∂ when ∂ belongs to a more general class of Cantor sets (with dim ∂ possibly below 1) [28] and then to Cantor repellers of any dimension [32], that is, sets K for which there are ⊆ C ⊆ C smooth disjoint domains Ui compactly contained in a domain U and univalent maps fi : Ui → U for which K is the unique compact set such that K = fi(U) (and in Volberg’s result, two of the maps need to be linear). Urba’nski and Zdunik have also shown that the attractors of conformal iterated function systems (IFS) have a dimension drop when either the limit set is contained in a real-analytic curve, if the IFS consists of similarities only, or if the IFS is irregular (see [31]). See also [27, 29]. A common thread to many of these results is etiher some uniform disconnectivity prop- erty (see equations (XI.2.1)-(XI.2.3) in [18, 22], [10, Lemma 2.5], and [15, Lemma 5]), or some self-similar or “dynamically defined” structure [28, 32, 33]. In the latter case, self- similarity allowed authors to exploit some ergodic theory, except the work of Batakis [10] which gave a non-ergodic proof of dimension drop for a wide class of Cantor sets that also works in higher dimensions, and later studied how the dimension is continuous with respect to the parameters defining the Cantor set [11, 12]. In the present paper, we develop a different sufficient condition for when the harmonic measure is strictly less than the dimension of the boundary. Though it assumes some strong conditions on the Hausdorff measure on the boundary, it requires no self-similar structure or uniform-disconnectedness. Instead, it assumes some uniform non-flatness condition (which will hold for any self-similar set in Rd+1 of dimension at least d that isn’t a d-dimensional plane), and also holds in higher dimensions.
Main Theorem Given d ∈ N, C1 > 0, and β>0, there are constants s0
Some remarks are in order. Firstly, this theorem does not cover all the fractals considered by Batakis, which may or may not satisfy (1.1). Secondly, nor does it tackle sets all sets Dimension Drop for Harmonic Measure on Ahlfors Regular Boundaries 1027 with dimension s
Theorem 1.1 [5, Theorem 1.1] Let d ≥ 1 and Rd+1 be an open connected set and let ω := ωp be the harmonic measure in where p is a fixed point in .LetE ⊂ ∂ be a subset with Hausdorff measure H d (E) < ∞. Then: d (a) If ω is absolutely continuous with respect to H on E,thenω|E is d-rectifiable, in the sense that ω-almost all of E can be covered by a countable union of n-dimensional Lipschitz graphs. (b) If H d is absolutely continuous with respect to ω on E,thenE is a d-rectifiable set, in the sense that H d -almost all of E can be covered by a countable union of d-dimensional Lipschitz graphs.
Thus, the boundary of our domain has tangents, but this violates (1.2). Using compactness arguments for harmonic measure is quite common, and the author first learned of it from the work of Kenig and Toro [23]. For more recent applications, see [21, Section 3], [13], and [7], and the references therein, which are primarily concerned with uniform domains. See also [8] for an example of compactness arguments used in non- uniform domains. We don’t know whether our result holds for all s ∈ (d −1,d+1].Ifs 2 Preliminaries We will let B(x,r) ={y :|x − y|≤r} and B = B(0, 1).IfB = B(x,r),we s s let λB = B(x,λr), xB = x,andrB = r. We will denote by H and H∞ the s- dimensional Hausdorff measure and Hausdorff content respectively. For a reference on geometric measure theory and Hausdorff measure, see [26]. We will write a b if there is a constant C>0sothata ≤ Cb and a t b if the constant depends on the parameter t. As usual we write a ∼ b and a ∼t b to mean a b a and a t b t a respectively. We will assume all implied constants depend on d and hence write ∼ instead of ∼d . Whenever A, B ⊂ Rd+1 we define dist(A, B) = inf{|x − y|; x ∈ A, y ∈ B}, and dist(x, A) = dist({x},A). Let diam A denote the diameter of A defined as diam A = sup{|x − y|; x,y ∈ A}. ∈ x For a domain and x ,weletω denote the harmonic measure for with pole at x and G(·, ·) the associated Green function. For a reference on harmonic measure and the Green function, see [4]. Given a domain and a ball B centered on ∂,wesayx is a c-corkscrew point for B ∩ if B(xB ,crB ) ⊆ B ∩ . We will say a domain has lower s-content regular complement with constant c1 if for all B centered on ∂ and 0 d+1 Lemma 2.1 If ⊆ R has lower s-content regular complement with constant c1 and s>d− 1, then there is b ∈ (0, 1) so that x ∈ ∈ ω(B(ξ, r)) c1 for ξ ∂, 0 A domain satisfies the capacity density condition (CDC) if, for all x ∈ ∂ and 0 < r Remark 2.2 We will state some lemmas below that assume the CDC, but keep in mind the CDC is implied by Eq. 2.1 for s>d− 1. This can be seen from [19, Lemma 2.31]). Dimension Drop for Harmonic Measure on Ahlfors Regular Boundaries 1029 Alternatively, Ancona showed in [9, Lemma 3] that the CDC is equivalent to the property that, for some α>0, | − | α x x xB ω ∩ (∂B ∩ ) for x ∈ B ∩ B r for all balls B centered on ∂ with rB < diam ∂. This property is implied by Bourgain’s lemma (see for example [6, Lemma 2.3]). In particular, the maximum principle implies the following lemma. Lemma 2.3 Let Rd+1 be an open set that satisfies the CDC and let x ∈ ∂.Then there is α>0 so that for all 0 Lemma 2.4 [1, Lemma 1] For x ∈ ⊆ Rd+1 and φ ∈ C∞(Rd+1), c x = + φω φ(x) φ(y)G(x, y)dy. (2.4) We will typically use the above lemma when φ(x) = 0. Lemma 2.5 Let ⊆ Rd+1 be a CDC domain. Let B be a ball centered on ∂ with 0 < rB < diam ∂.Then ωx (4B) G (x, y) for all x ∈ \ B and y ∈ B d−1 2 (2.5) rB The above lemma has evolved over many years and this isn’t the most general statement, but it will suit our purposes. It follows from the proof of Lemma 3.5 in [2]: it is assumed there that the domain is John (and so in particular bounded) but is not necessary for the above statement. A version of this is also shown in [5] that works for general bounded domains without the CDC (and implies the above inequality), but it is only for d>1. Recall that a Harnack chain between two points x,y ∈ is a sequence of balls B1, ...,Bn for which 2Bi ⊆ . By Harnack’s inequality, there is M>0 so that for any non-negative harmonic function u on , u(x) ≤ Mnu(y). (2.6) d+1 Lemma 2.6 [8, Lemma 2.9] Let j ⊂ R be a sequence of domains with lower s-content regular complements, s>d− 1, 0 ∈ ∂j , inf diam ∂j > 0, and suppose there is a ball x0 B0 = B(x0,r0) ⊂ j for all j ≥ 1. Then there is a connected open set ∞ containing B so that, after passing to a subsequence, c (1) G (x , ·) → G x (x , ·) uniformly on compact subsets of {x } , j 0 ∞0 0 0 x0 x0 (2) ω ω x , and j ∞0 x (3) ∞0 also has lower s-content regular complement with the same constant. This is not how it is stated in [8], but it follows from the proof. Indeed, the lemma is stated for CDC domains, but they use the fact that CDC domains have lower s-content regular 1030 J. Azzam complements for a particular dimension and constant, and then use that characterization to prove the lemma. 3 Rectifiability Definition 3.1 AsetE ⊆ Rn is d-rectifiable if H d (E) < ∞ anditmaybecoveredupto H d -measure zero by a countable union of rotated Lipschitz graphs. For two measures μ and ν and a ball B,let FB (μ, ν) = sup fdμ− fdν f where the supremum is over all nonnegative 1-Lipschitz functions supported in B. For x ∈ Rn, r>0, μ a Radon measure, and an affine d-dimensional plane V ,let −d−1 d αμ(x, r, V ) = r inf FB(x,r)(μ, cH |V ) c>0 and let αμ(x, r) = inf αμ(x, r, V ) V where the infimum is over all d-dimensional affine planes V . Lemma 3.2 [30, Lemma 2.1] If μ is a Radon measure and is a Lipschitz graph, then 1 2 dr d αμ(x, r) < ∞ for H -a.e. x ∈ . (3.1) 0 r := d As a corollary, if we let αE αH |E , we get the following. Corollary 3.3 If E is a Borel set and F ⊆ E is d-rectifiable, then d lim αE(x, r) = 0 for H -a.e. x ∈ F . r→0 This follows from Lemma 3.2 since F may be covered by countably many d-dimensional d+1 Lipschitz graphs and since αμ(x, r) ≤ (s/r) αμ(x, s) for r Remark 3.4 The proof of the corollary is much simpler than envoking Lemma 3.2, and is quite standard, but we couldn’t find a short reference for it. It can actually be proven more simply by the techniques in Chapters 14-16 of [26], although for the sake of brevity we didn’t want to recall too much background in order to do this. 4 Non-Flatness or Big Dimension Implies Change in Density For a Radon measure μ, s ≥ 0, and a ball B,wedefine μ(B) s (B) = . μ rs d+1 Throughout this section, we will work with a domain ⊆ R and we will let ω = ω for short. Dimension Drop for Harmonic Measure on Ahlfors Regular Boundaries 1031 d+1 Lemma 4.1 Suppose s>d∈ N, c, C1,M > 0, and ⊆ R has lower s-content −1 regular with constant C1.LetB be centered on ∂, A>4, A0 > 0, and x ∈ B\2A B a c-corkscrew point in B (meaning x ∈ B ∩ and B(x,2crB ) ⊆ ) so that x −1 ω (A B) ≥ A0. −1 Then there is δ = δ(d,c,M,C1,A0,A)> 0 and a ball B ⊆ 5A B with rB ≥ δrB so that s −s ωx (B )>MrB Proof Without loss of generality, B = B. We first use a method of Aikawa and Hirata [2] to show there is a Harnack chain from x to a point in 2A−1B.Letφ be a smooth function supported on 2A−1B and equal to 1 on A−1B.Then x (2.4) A0 ≤ φdω = G(x,y) φ(y)dy sup G(x,y). y∈2A−1B Thus, there is a universal constant λ>0sothat y ∈ Eλ ={z : G(x, z) > λA0}. 1−d Note that as G(x, z) |x − z| , Eλ is a bounded set with diameter depending on λ. Moreover, Eλ is open and also contains x. By the maximum principle, it must also be −1 connected, so there is a curve γ ⊆ Eλ joining x to a point in 2A B. Note that if y ∈ γ δ(y) < ε,andξ ∈ ∂ is closest to y,thenx ∈ B(ξ,c/2),andso (2.3) (2.5) α x 1−d α G(x,y) ||G(x, ·)||L∞(B(ξ,c/4))ε ω (B(ξ, c))c ε and so y ∈ Eλ for ε small enough (depending on A and d). Thus, dist(y, ∂) ≥ ε>0for all y ∈ γ . Thus, we can find a Harnack chain from x to a point y ∈ γ ∩2A−1B of uniformly bounded length (depending on ε, d, and A). = y−ξ = Again, let ξ be the closest point in ∂ to y.Letv |y−ξ| and for t>0letBt B(ξ,t) (see Fig. 1). Recall that if d>1, for any r>0andz ∈ B(y,r)\{y}, 1−d 1−d GB(y,r)(y, z) = c(|y − z| − r ) where c>0 is some constant depending on d. (If d = 1, Green’s function is instead a r multiple of log |y−z| , and the estimates below are similar in this case). Thus, for t>0 small, and if d>1, (2.5) x d−1 (2.6) d−1 ω (4Bt ) G(x, ξ + tv)t ∼ ε G(y, ξ + tv)t ≥ G (y, ξ + tv)td−1 B(y,δ(y)) = c |y − (ξ + tv)|1−d − δ (y)1−d td−1 1−d 1−d d−1 = c (δ(y) − t) − δ(y) t − = − 1−d − td 1 td c (1 t/δ(y)) 1 d−1 d . δ(y) δ(y) Note that as there is a Harnack chain of length depending on ε between x and y,and because x is a c-corkscrew point in B,wehavethat δ(y) ∼ε δ(x) ∼c rB = 1. x d In particular, for s>d, the above estimates imply ω (4Bt ) t . In particular, s d−s s ∈ −1 ωx (4Bt ) t , hence for t small (depending on M), ωx (4Bt )>M.Sincey 2A B 1032 J. Azzam and the center of B is in ∂, |ξ − y|≤2A−1,sofort small enough, we can also guarantee −1 that 4Bt ⊆ 5A B,andso4Bt is our desired ball. Lemma 4.2 Given d ∈ N, M,c,C1 > 0, and β>0,thereiss0 x −1 ◦ ω (A B ) ≥ A0. −1 Then there is δ = δ(d,c,C1,M,β,A0,A)> 0 and a ball B ⊆ 5A B so that rB ≥ δrB and s −s ωx (B )>MrB . Proof Again, we can assume B = B. Suppose instead that for all j ∈ N we could find a − 1 ≤ + domain j with C1-Ahlfors sj -regular boundary with d j x − ◦ ω j (A 1B ) ≥ A j 0 ⊆ 1 B ≥ 1 yet for all B 2 with rB j , x s ω j (B) ≤ Mr j . j B Since the ∂j are Ahlfors sj -regular with constant C1 and sj → s,forj large they all uniformly have large (d − 1/2)-content regular complements. By Lemma 2.6, we may pass −1 to a subsequence so that sj → s ∈[d,d + 1], xj → x0 ∈ B\2A B a c-corkscrew point in x xj x B for a domain = ∞0 so that ω ω 0 and also has lower (d − 1/2)-content regular 0 j 0 complement. In particular, one can show that ωx0 (B) ≤ Mrd (4.1) 0 B −1 for all balls B centered on ∂0 contained in 5A B,and − ◦ ωx0 (A 1B ) ≥ A . (4.2) 0 0 Hence, by the previous lemma, Eq. 4.1 is impossible if s>d, thus we must have s = d. In particular, ωx0 H d in 5A−1B. 0 Since the ∂j are Ahlfors sj -regular with constant C1 and sj → s, we can also pass to a subsequence so that ∂j ∩ 10B converges in the Hausdorff metric to a set so that if d s σ = H |,thenσ(B) ∼ r for all B ⊆ 10B centered on . We will now show that for this new set we have bβ(x, r) ≥ β>0forallx ∈ ∩ B, 0 Fig. 1 The balls 4Bt and B(y,δ(y)). Observe that B(y,δ(y) is tangent to ∂ If zj ∈ B(x,r) ∩ ∂j is farthest from P and yj ∈ P ∩ B(x,r) is farthest from ∂j ,then we can pass to a subsequence so that zj → z ∈ B(x,r) ∩ and yj → y ∈ P ∩ B(x,r), and then ≤ + β lim supj→∞ supp∈∂ ∩B(x,r) dist(p, P ) supq∈P ∩B(x,r) dist(q, ∂j ) j = + lim supj→∞ dist(zj ,P) dist(yj ,∂j ) = dist(z, P ) + dist(y, ) < β, which is a contradiction, and this proves (4.3). −1 d −1 Thus, ∂0 ∩ 5A B ⊆ ,soH (∂0 ∩ 5A B)<∞.ByEqs.4.1 and 4.2, and because ωx0 H d on ∂ ∩ 5A−1B,thereisE ⊆ ∂ ∩ 5A−1B with 0 < 0 0 0 H d (E) < ∞ so that H d ωx0 H d on E. To see this, just observe that if 0 x0 d f = dω | −1 /dH | −1 E ={x ∈ 5A B∩∂0 5A B∩∂0 is the Radon-Nikodym derivative, then −1 5A B ∩ ∂0 : f(x)> 0} is our desired set (see [26, Theorem 2.12]). Now Theorem 1.1 implies that E is a d-rectifiable set of positive H d -measure. Since E ⊆ , Corollary 3.3 implies that lim α(x, r) = 0. r→0 for some x ∈ E. In particular, if ε>0, there is r>0 small enough and a plane V so that α(x, 4r, V ) < . (4.4) 1034 J. Azzam Without loss of generality, we can assume V ∩ B(x,4r) =∅. Indeed, suppose instead that V ∩ B(x,4r)∅.Letφ(z) = (4r −|x − z|)+,thenφ(z) = 0onV and φ ≥ 2r on B(x,2r). This and the fact that is Ahlfors regular imply −d−1 ε>α(x, 4r, V ) ≥ (4r) φ(z)dσ(z) ≥ −d−1 (4r) σ(B(x,2r))2r C1 1. (4.5) which is impossible for ε small enough depending on C1,andsoV ∩ B(x,r) =∅for ε>0 small. But by Eq. 4.3,forallr>0sothatB(x,r) ⊆ B (since x ∈ 5A−1B), there is either y ∈ B(x,r)∩ so that dist(y, V ) ≥ δr,orthereisy ∈ V ∩B(x,r) so that dist(y, ) ≥ δr. ≤ ≤ 1 In the former case, if we let 0 φ 1bea 2r -Lipschitz function equal to 1 on B(x,2r) and zero outside B(x,4r),thenψ(z) = φ(z)dist(z, V ) is a 6-Lipschitz function (recall dist(z, V ) ≤ 8r for all z ∈ B(x,4r) since V ∩ B(x,4r) =∅)andψ(z) ≥ δr/2on B(y,δr/2). Hence, −1 d+1 −d ≤ −d ≤ −d−1 (4.4) C1 δ 2 r δσ(B(y, rδ/2)) 2r ψdσ < ε, which is a contradiction for ε small enough. The case that there is y ∈ V ∩ B(x,r) so that dist(y, ) ≥ δr has a similar proof and we omit it. Lemma 4.3 Given d ∈ N, C1 > 0, and β>0,thereiss0 (2.3) ≤ α ˜ p B ∩ + x 1 B ˜ p Cε ω (∂ ) E ω 10 dω (x). By Lemma 2.1, ωx(aB) 1on∂B ∩ , so by the maximum principle, (4.6) 1 ω˜ p(∂B ∩ ) ωp(aB) ≤ M asωp( B). (4.8) 1 10 1This follows from the Brownian motion definition of harmonic measure, but for a direct proof, see the appendix in [3]. Dimension Drop for Harmonic Measure on Ahlfors Regular Boundaries 1035 So for ε>0 small enough depending on M1 and a,byEqs.4.7 and 4.8, 1 1 ωp B ≤ 2 ωx B dω˜ p(x). (4.9) 10 E 10 Let Bj be a covering of E by boundedly many balls centered on E of radius ε/4 (whose total number depends only on ε and d), so 2Bj ⊆ . We claim we there are t>0 (depending on ε and M1)andj so that if B = Bj ,then 1 1 ω˜ p(B ) ≥ tωp B and ωxB B ≥ t. 10 10 ˜ p p 1 B If not, then for each j either ω (Bj ) Thus, we have a ball B centered on ∂B with 2B ⊆ and rB = ε/2, and such that 1 (4.6) − − ω˜ p(B ) ωp B ≥ M 1a sωp(aB). (4.10) ε,M1 10 1 xB 1 B ≥ ⊆ B ⊆ B ε and ω ( 10 ) t.LetM>0. Now since B a , xB is a 2a -corkscrew point for aB, and so Lemma 4.2 (applied with aB in place of B, A = 10a, asM in place of M,and = ⊆ −1 B = 1 B A0 t) implies there is δ>0 depending on M,d,C1,andβ and B 5A (a ) 2 s s with r ≥ δ so that x (B) > M. By Harnack’s inequality, x (B) M for all x ∈ B . B ω B ω Thus, ωp(B) = ωx(B)dω˜ p(x) ≥ ωx(B)dω˜ p(x) ∂B∩ B ∩∂B (4.10) ω˜ p(B )Mrs ωp(aB)Mrs . B ε,M1,a B Thus, for M large enough (depending on ε and M1), we have s = p −s −s p B = s B ωp (B) ω (B)rB >M1a ω (a ) M1ωp (a ), which proves the lemma. 5 Proof of the Main Theorem d+1 Let C1,β,s,d,β,and ⊆ R be as in the Main Theorem where s0 is the constant from = p ∈ Lemma 4.3, and set ω ω for some p . We recall the following version of “dyadic cubes” for metric spaces, first introduced by David [17] but generalized in [16]and[20]. 1036 J. Azzam Theorem 5.1 Let X be a doubling metric space. Let Xk be a nested sequence of maximal k ρ -nets for X where ρ<1/1000 and let c0 = 1/500.Foreachn ∈ Z there is a collection Dk of “cubes,” which are Borel subsets of X such that the following hold. = (1) For every integerk, X Q∈D Q. k (2) If Q, Q ∈ D = Dk and Q ∩ Q =∅,thenQ ⊆ Q or Q ⊆ Q. k(Q) (3) For Q ∈ D,letk(Q) be the unique integer so that Q ∈ Dk and set (Q) = 5ρ . Then there is ζQ ∈ Xk so that BX(ζQ,c0(Q)) ⊆ Q ⊆ BX(ζQ, (Q)) (5.1) and Xk ={ζQ : Q ∈ Dk}. D ∈ ∈ D Let be the Christ-David cubes for ∂.Fixn0 so that x0 aBQ for all Q n0 .By rescaling, we can assume without loss of generality that n0 = 0. Let ω(Q) s (Q) = . ω σ(Q) ≥ ∈ D ∈ N Lemma 5.2 Let n 0, M2 > 0, and Q n.ThereisNQ so that NQ M2,C1,β,d 1 c ∈ D + 0 and such that there is Q n NQ contained in 2 BQ so that s ∈[ −1 s s ] ω(Q ) M2 ω(Q), M2ω(Q) . ˜ Proof Fix N ∈ N large enough so that if Q ∈ Dn+N is the cube with same center as Q,then c0 aB ˜ ⊆ B (5.2) Q 2 Q where a is as in Lemma 4.3. Since N is fixed and only depends on some fixed universal constants, we will not indicate when constants depend on it below. Clearly we can assume s ˜ ≥ −1 ω(Q) M2 ω(Q), (5.3) otherwise we’d pick Q = Q˜ . Thus, (5.1) (5.3) (5.2) s s ˜ −1 s −1 s (B ˜ ) (Q) ≥ M (Q) M (aB ˜ ), (5.4) ω Q C1 ω 2 ω N,C1 2 ω Q so by Lemma 4.3, for M1 > 0thereisδ depending on M1,M2,d,C1,β and N,andthere ⊆ 1 ˜ ≥ ˜ is B 2 BQ˜ centered on ∂ (and thus also centered on Q) with rB δ(Q) for which s ∈[ −1 s s ] ω(B) M1 ω(aBQ˜ ), M1ω(aBQ˜ ) . s −1 s Suppose first that ω(B) < M1 ω(aBQ˜ ).ThenwetakeQ to be the largest cube containing the center of B that is also contained in B,so ˜ (Q ) ∼ rB δ(Q) (5.5) Furthermore, (5.1) (5.5) (5.4) s s −1 s M2 s ˜ (Q ) (B) < M (aB ˜ ) (Q) ω C1 ω 1 ω Q C1 ω M1 2 and the lemma follows in this case by picking M1 M2 . s ≥ s Now suppose that ω(B) M1ω(aBQ˜ ).LetN be the largest integer for which N ˜ 5ρ <(Q)/4. Then there are at most boundedly many cubes in DN which cover Dimension Drop for Harmonic Measure on Ahlfors Regular Boundaries 1037 B ∩ ∂, and one of them, call it Q ,musthaveω(Q ) ω(B).SinceQ ∩ B =∅and = N ˜ ⊆ 1 (Q ) 5ρ <(Q)/4, and B 2 BQ˜ ,wehave (5.1) c0 Q ⊆ B ˜ ⊆ B . Q 2 Q ˜ Also, we have (Q ) ∼δ (Q) ∼N (Q) and (5.4) s s ≥ s s ˜ ≥ M1 s ω(Q ) ω(B) M1ω(aBQ˜ ) C1 M1ω(Q) ω(Q). M2 2 Again, the lemma follows by picking M1 M2 . In either case, since δ always depends on d,β,C1,M1 and M2 (and because M1 depends on d,β,C1, and M2), we have ∈ D (Q ) d,β,C1,M2 (Q),soifNQ is such that Q n+NQ ,thenNQ d,β,C1,M2 1, and we’re done. We now let ε>0 be small and let NQ denote the integer from the previous lemma −1 applied when M2 = ε . Fix a cube Q0 ∈ D0 and define families of sub-cubes of Q0 in D D ={ } D ∈ D n inductively as follows. First let 0 Q0 ,thenif n has been defined and R n, ∈ D and nR is so that R nR ,let ={ ∈ D : ⊆ } Ch(R) Q nR+NR Q R and D = D = D n+1 Ch(R), n. ∈D R n Lemma 5.3 There is λ = λ(d, C1,β)∈ (0, 1) so that for all R ∈ D , 1 1 1 1 ω(Q)2 σ(Q)2 <λω(R)2 σ(R)2 . (5.6) Q∈Ch(R) Proof Let R be the cube obtained in Lemma 5.2 applied to Q = R with M2 = 16. Suppose first that 1 s (R )< s (R). (5.7) ω 16 ω By the Cauchy-Schwartz inequality, 1 1 1 1 2 2 2 2 ≤ ω(Q) σ(Q) Q∈Ch(R) ω(Q) Q∈Ch(R) σ(Q) Q =R Q∈Ch(R) Q =R 1 1 = ω(R)2 (σ(R) − σ(R )) 2 . Also, by Eq. 5.7, 1 1 1 1 1 2 2 2 2 2 = ω(R ) 1 ω(R) ω(R ) σ(R ) σ(R ) σ(R )< 16 σ(R) σ(R ) 1 1 σ(R ) = 1 2 2 4 ω(R) σ(R) σ(R) . 1038 J. Azzam 1 − 2 ≤ − x ≤ These two estimates and the fact that (1 x) 1 2 for all x 1 imply that for ε small, 1 1 1 1 1 1 σ(R ) 2 2 ≤ 2 − 2 + 1 2 2 ω(Q) σ(Q) ω(R) (σ(R) σ(R )) 4 ω(R) σ(R) σ(R) Q∈Ch(R) 1 1 1 σ(R ) 2 σ(R ) = 2 2 − + 1 ω(R) σ(R) 1 σ(R) 4 σ(R) 1 1 σ(R ) ≤ 2 2 − 1 ω(R) σ(R) 1 4 σ(R) ∈ D Now (5.6) follows since, for R n,sinceNR β,C1,d 1, we have σ(R ) (R )s (R ) s (R ) d = ≥ 1. (5.8) σ(R) C1 (R)s (R) (R) C1,β,d Hence, there is t = t(C1,β,d)>0sothat 1 σ(R ) t 1 − ≤ 1 − =: λ<1 4 σ(R) 4 and the lemma follows in this case. Now suppose that s s ω(R )>16ω(R). (5.9) By the Cauchy-Schwartz inequality, 1 2 1 1 1 2 2 2 ≤ ω(Q) σ(Q) Q∈Ch(R) ω(Q) Q∈Ch(R) σ(Q) Q =R Q∈Ch(R) Q =R 1 1 = (ω(R) − ω(R )) 2 σ(R)2 . By Eq. 5.9, 1 1 1 1 1 2 2 2 2 2 = σ(R ) 1 σ(R) ω(R ) σ(R ) ω(R ) ω(R ) < 16 ω(R ) ω(R) 1 1 ω(R ) = 1 2 2 4 ω(R) σ(R) ω(R) . Just as earlier, we have 1 1 1 1 1 1 ω(R ) 2 2 ≤ − 2 2 + 1 2 2 ω(Q) σ(Q) (ω(R) ω(R )) σ(R) 4 ω(R) σ(R) ω(R) Q∈Ch(R) 1 1 1 ω(R ) 2 ω(R ) ≤ 2 2 − + 1 ω(R) σ(R) 1 ω(R) 4 ω(R) 1 1 ω(R ) ≤ 2 2 − 1 ω(R) σ(R) 1 4 ω(R) Now we use the fact that s (5.9) ω(R ) (5.8∼) ω(R ) ≥ C1,β,d s 1. ω(R) ω(R) = ω(R ) Thus, there is t t(C1,β,d)>0sothat ω(R) >t. Hence, we again have 1 ω(R ) t 1 − ≤ 1 − =: λ<1 4 ω(R) 4 and again the lemma follows. Dimension Drop for Harmonic Measure on Ahlfors Regular Boundaries 1039 Let R ∈ D . Then for Q ∈ Ch(R) and for some c = c(β, d, C1), σ(Q) ≥ cσ(R). Let τ ∈ (0, 1) be small, we will fix its value later. Then 1 1−τ − τ − τ 1 1 2 2 ≤ ω(Q) σ(Q) σ(R) 2 c 2 Q∈Ch(R) ω(Q)2 σ(Q)2 Q∈Ch(R) (5.6) − τ 1 1−τ 1 1−τ Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 Inter- national License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. 1040 J. Azzam References 1. Aikawa, H.: Equivalence between the boundary Harnack principle and the Carleson estimate. Math. Scand. 103(1), 61–76 (2008) 2. Aikawa, H., Hirata, K.: Doubling conditions for harmonic measure in John domains. Ann. Inst Fourier (Grenoble) 58(2), 429–445 (2008) 3. Akman, M., Azzam, J., Mourgoglou, M.: Absolute continuity of harmonic measure for domains with lower regular boundaries. arXiv:1605.07291 (2016) 4. Armitage, D.H., Gardiner, S.J.: Classical Potential Theory. Springer Monographs in Mathematics. Springer-Verlag London Ltd., London (2001) 5. Azzam, J., Hofmann, S., Martell, J.M., Mayboroda, S., Mourgoglou, M., Tolsa, X., Volberg, A.: Rectifiability of harmonic measure. Geom. Funct Anal. 26(3), 703–728 (2016) 6. Azzam, J., Mourgoglou, M.: Tangent measures and absolute continuity of harmonic measure. Rev. Mat. Iberoam. 34(1), 305–330 (2018) 7. Azzam, J., Mourgoglou, M.: Tangent measures and densities of harmonic measure. To appear in Rev Math. (2015) 8. Azzam, J., Mourgoglou, M., Tolsa, X.: A two-phase free boundary problem for harmonic measure and uniform rectifiability. arXiv:1710.10111 (2017) 9. Ancona, A.: On strong barriers and an inequality of Hardy for domains in Rn. J. London Math. Soc. (2) 34(2), 274–290 (1986) 10. Batakis, A.: Harmonic measure of some Cantor type sets. Ann. Acad. Sci. Fenn. Math. 21(2), 255–270 (1996) 11. Batakis, A.: A continuity property of the dimension of the harmonic measure of Cantor sets under perturbations. Ann. Inst. H. Poincare´ Probab. Statist. 36(1), 87–107 (2000) 12. Batakis, A.: Dimension of the harmonic measure of non-homogeneous Cantor sets. Ann. Inst. Fourier (Grenoble) 56(6), 1617–1631 (2006) 13. Bortz, S., Engelstein, M.: Reifenberg flatness and oscillation of the unit normal vector. arXiv:1708.05331 (2017) 14. Bourgain, J.: On the Hausdorff dimension of harmonic measure in higher dimension. Invent. Math. 87(3), 477–483 (1987) 15. Carleson, L.: On the support of harmonic measure for sets of Cantor type. Ann. Acad. Sci. Fenn. Ser. A IMath.10, 113–123 (1985) 16. Christ, M.: A T(b)theorem with remarks on analytic capacity and the Cauchy integral. Colloq. Math. 60/61(2), 601–628 (1990) 17. David, G.: Morceaux de graphes lipschitziens et integrales´ singulieres` sur une surface. Rev. Mat. Iberoamericana 4(1), 73–114 (1988) 18. Garnett, J.B., Marshall, D.E.: Harmonic Measure, Volume 2 of New Mathematical Monographs. Cambridge University Press, Cambridge (2008). Reprint of the 2005 original 19. Heinonen, J., Kilpelainen,¨ T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Dover Publications, Inc., Mineola (2006). Unabridged republication of the 1993 original 20. Hytonen,¨ T., Martikainen, H.: Non-homogeneous Tb theorem and random dyadic cubes on metric measure spaces. J. Geom. Anal. 22(4), 1071–1107 (2012) 21. Hofmann, S., Martell, J.-M., Mayboroda, S., Toro, T., Zhao, Z.: Uniform rectifiability and elliptic operators with small carleson norm. arXiv:1710.06157 (2017) 22. Jones, P.W., Wolff, T.H.: Hausdorff dimension of harmonic measures in the plane. Acta Math. 161(1–2), 131–144 (1988) 23. Kenig, C.E., Toro, T.: Free boundary regularity for harmonic measures and Poisson kernels. Ann. Math. (2) 150(2), 369–454 (1999) 24. Lewis, J.L., Verchota, G.C., Vogel, A.L.: Wolff snowflakes. J. Math. 218(1), 139–166 (2005) 25. Makarov, N.G.: On the distortion of boundary sets under conformal mappings. Proc. London. Math. Soc. (3) 51(2), 369–384 (1985) 26. Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces, Volume 44 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1995). Fractals and rectifiability 27. Meyer, D.: Dimension of elliptic harmonic measure of snowspheres. Illinois J. Math. 53(2), 691–721 (2009) 28. Makarov, N., Volberg, A.: On the harmonic measure of discontinuous fractals (1986) 29. Popovici, I.: Rigidity and Dimension of the Harmonic Measure on Julia Sets. ProQuest LLC, Ann Arbor (1998). Thesis (Ph.D.)–Michigan State University 30. Tolsa, X.: Uniform measures and uniform rectifiability. J. Lond. Math. Soc. (2) 92(1), 1–18 (2015) Dimension Drop for Harmonic Measure on Ahlfors Regular Boundaries 1041 31. Urbanski,´ M., Zdunik, A.: Hausdorff dimension of harmonic measure for self-conformal sets. Adv. Math. 171(1), 1–58 (2002) 32. Volberg, A.L.: On the harmonic measure of self-similar sets on the plane. In: Harmonic Analysis and Discrete Potential Theory, pp. 267–280. Springer (1992) 33. Volberg, A.L.: On the dimension of harmonic measure of Cantor repellers. Michigan Math. J. 40(2), 239–258 (1993) 34. Wolff, T.H.: Counterexamples with harmonic gradients in R3. In: Essays on Fourier Analysis in Honor of Elias M. Stein (Princeton, NJ, 1991), Volume 42 of Princeton Math. Ser., pp. 321–384. Princeton Univ. Press, Princeton (1995) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.