Geometric Measure Theory–Recent Applications

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Geometric Measure Theory Recent Applications

Tatiana Toro

GMT Introduction questions that gave rise to the field, briefly mention some Geometric Measure Theory (GMT) provides a framework of the milestones, and then focus on some of the recent to address questions in very different areas of mathemat- developments at the intersection of GMT, potential theory, ics, including calculus of variations, geometric analysis, po- and harmonic analysis. tential theory, free boundary regularity, harmonic analy- The origins of the field can be traced to the following sis, and theoretical computer science. Progress in different question: do the infinitesimal properties of a measure deter- branches of GMT has led to the emergence of new chal- mine the structure of its support? lenges, making it a very vibrant area of research. In this In the late 1920s and early ’30s Besicovitch was inter- 2 note we will provide a historic background to some of the ested in understanding the structure of a set 퐸 ⊂ ℝ satisfying 0 < ℋ1(퐸) < ∞ and such that for ℋ1-a.e. 푥 ∈ 퐸, Tatiana Toro is the Craig McKibben & Sarah Merner Professor in Mathematics ℋ1(퐵(푥, 푟) ∩ 퐸) at the University of Washington. She was partially supported by NSF grant lim = 1, (1) + DMS-1664867. Her email address is [email protected]. 푟→0 2푟 Communicated by Notices Associate Editor Chikako Mese. where ℋ1 denotes the 1-dimensional Hausdorff measure. For permission to reprint this article, please contact: (The formulation above is a modern version of the prob- [email protected]. lem. Besicovitch, who most likely was unaware of the exis- DOI: https://doi.org/10.1090/noti1853 tence of the Hausdorff measure, formulated the question

474 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 4 in terms of the linear measure.) For 푛 ≥ 1, the 푛-dimen- 휇 = ℋ푛 퐸 by a general Radon measure 휇? Initially 푛 푚 sional Hausdorff measure ℋ in ℝ generalizes the no- progress on these questions was slow. tions of length of a curve (푛 = 1), surface area (푛 = In 1944, Morse and Randolph [52] proved when 푚 = 푚 2), and volume (푛 = 3) to subsets of ℝ . Moreover 2 that if 휇 is a Radon measure on ℝ푚 for which the 1- 푛 푛 푛 푛 ℒ ℝ = ℋ ℝ density exists 휇-a.e., then 휇 is 1-rectifiable. In 1950, Moore Besicovitch showed that if 퐸 satisfies the hypothesis (1), [51] showed that this result holds for any 푚. In 1961, then 퐸 is 1-rectifiable; that is, 퐸 is contained in a countable Marstrand [48] showed that if 퐸 ⊂ ℝ3 and the 2-density union of Lipschitz images of ℝ union a set of 1-Hausdorff exists for ℋ2 퐸, ℋ2-a.e 푥 ∈ ℝ3, then 퐸 is 2-rectifiable. measure 0 (see [16], [17]). In GMT the notion of rectifia- In 1975, Mattila [49] proved that if 퐸 ⊂ ℝ푚 and the 푛- bility is used to describe the structure (also the regularity) density exists for ℋ푛 퐸, ℋ푛-a.e 푥 ∈ ℝ푚, then 퐸 is 푛- of a set or a measure in a way similar to how the degree of rectifiable, completing the study of the problem formea- differentiability of charts is used to describe the smooth- sures that were defined as the restriction of Hausdorff mea- 푚 ness of a manifold in differential geometry. A set 퐸 ⊂ ℝ sure to a subset of Euclidean space. This still left open the is 푛-rectifiable if case of a general Radon measure. ∞ In 1987, in a true masterpiece, Preiss [56] showed that 푛 푚 퐸 ⊂ ⋃ 푓푗(ℝ ) ∪ 퐸0, if 휇 is a Radon measure on ℝ for which the 푛-density 푖=1 exists for 휇-a.e 푥 ∈ ℝ푚, then 휇 is 푛-rectifiable. Preiss 푛 푚 푛 where 푓푗 ∶ ℝ → ℝ is a Lipschitz map and ℋ (퐸0) = introduced a number of new tools and ideas whose appli- 0. Recall that 푓푗 is Lipschitz if there exists 퐿푗 > 0 s.t. for cations are still being unraveled and play a central role in 푥, 푦 ∈ ℝ푛 the results to be discussed later in this article. The question of rectifiability of a measure carries information about the |푓푗(푥) − 푓푗(푦)| ≤ 퐿푗|푥 − 푦|. fine structure of its measure-theoretic support. Motivated In 1947, Federer [30] proved a general converse of Besicov- by this perspective, Preiss introduced the notion of tangent 푚 itch’s theorem: if 푛 < 푚 and 퐸 ⊂ ℝ is 푛-rectifiable, then measures, which play the role that derivatives do when an- 푛 for ℋ -a.e. 푥 ∈ 퐸 alyzing the regularity of a function. They are obtained by ℋ푛(퐵(푥, 푟) ∩ 퐸) a limiting process of rescaled multiples of the initial mea- lim = 1, 푛 (2) 푟→0+ 휔푛푟 sure. Preiss’s argument includes a number of major steps, some of which have given rise to very interesting questions. where 휔푛 denotes the Lebesgue measure of the unit ball in ℝ푛. Roughly speaking, a blow up procedure shows that when 푛 휇 휇 휇 We introduce some terminology that will help us set the -density of exists -a.e., then at -a.e. point all tan- 푚 푛 휈 푛 the framework. Let 휇 be a Radon measure in ℝ (i.e. a gent measures are -uniform. A measure is -uniform 퐶 > 0 푟 > 0 푥 Borel regular measure that is finite on compact sets). The if there is a constant so that for and in Λ = spt 휈 = {푦 ∈ ℝ푚 ∶ 휈(퐵(푦, 푠)) > 푛-density of 휇 at 푥 the support 0 for every 푠 > 0} of 휈, we have 휇(퐵(푥, 푟)) 휃푛(휇, 푥) ∶= lim 푛 (3) 푛 푟→0 휔푛푟 휈(퐵(푥, 푟)) = 퐶푟 . (4) exists if the limit exists and 휃푛(휇, 푥) ∈ (0, ∞) 푚 His argument now requires a detailed understanding of A locally finite measure 휇 on ℝ is 푛-rectifiable if 휇 is the structure and geometry of the support of 푛-uniform absolutely continuous with respect to ℋ푛 (휇 ≪ ℋ푛, i.e 푛 measures. By work of Kirchheim and Preiss [43], the sup- ℋ (퐹) = 0 implies 휇(퐹) = 0) and port of an 푛-uniform measure 휈 is an analytic variety. Thus, ∞ using the fact that at 휇-a.e point, tangent measures to tan- 푚 푛 휇(ℝ \ ⋃ 푓푗(ℝ )) = 0, gent measures to 휇 are tangent measures to 휇, he showed 푗=1 that at 휇-a.e. point there is an 푛-flat tangent measure; that 푛 푚 where each 푓푗 ∶ ℝ → ℝ is Lipschitz. Recasting the re- is, a measure that is a multiple of the 푛-dimensional Haus- sults above in this light, we have in the case when 푚 = 2 dorff measure restricted to an 푛-plane. Then he showed and 푛 = 1, that by Besicovitch’s work if 퐸 ⊂ ℝ푚 such that that either an 푛-uniform measure 휈 is an 푛-flat measure 0 < ℋ푛(퐸) < ∞, and for ℋ푛-a.e. 푥 ∈ 퐸 the density of or its support is very far away from any 푛-plane. Using a 휇 = ℋ푛 퐸 exists and is 1, then 퐸 is 푛-rectifiable. By Fed- deep result about the “cones” of measures, he proves that erer’s work the converse in any dimension is true, that is, necessarily at 휇-a.e point all tangent measures are 푛-flat. if 퐸 ⊂ ℝ푚 is 푛-rectifiable then the density of 휇 = ℋ푛 퐸 Then modulo showing that this implies that the measure- exists and is 1 for ℋ푛-a.e. 푥 ∈ 퐸. Two natural questions theoretic support of 휇 satisfies the hypothesis of the Mar- arise at this point: 1) does Besicovitch’s result hold for any strand-Mattila rectifiability criterion, one concludes that 휇 푛, 푚 ∈ ℕ with 푛 < 푚? 2) what happens if we replace is 푛-rectifiable.

APRIL 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 475 Harmonic Measure To illustrate how ideas from one field can have a profound impact in another, we will focus on some of the recent applications of Preiss’s work to harmonic analysis and potential theory. The harmonic measure is a canonical measure associated to the Laplacian (see definition below). It plays a fundamental role in potential theory, constitutes the main building block for the solutions of the classical Dirichlet problem, and in non-smooth domains is the ob- ject that allows us to describe boundary regularity of the solutions to Laplace’s equation. We recall some of the background. Let Ω ⊂ ℝ푛+1 be a bounded domain, let 푓 be a continuous function on the boundary of Ω, i.e. 푓 ∈ 퐶(휕Ω). The classical Dirichlet problems asks whether there exists a function 푢 ∈ 퐶(Ω) ∩ 푊1,2(Ω) such that

Δ푢 = 0 in Ω { (5) 푢 = 푓 on 휕Ω.

1,2 Figure 1. Kowalski-Preiss cone Here 푢 ∈ 푊 (Ω) means that 푢 and its weak derivatives are in 퐿2(Ω) and Δ푢 = 0 is interpreted in the weak 휁 ∈ 퐶1(Ω) A natural and easy-to-state question derived from this sense; that is, for any 푐 , work is: what does the support of an 푛-uniform measure on ℝ푚 really looks like? Clearly the restriction of 푛-Haus- ∫⟨∇푢, ∇휁⟩ = 0. dorff measure to an 푛-plane is 푛-uniform, but are there other examples? Work of Preiss [56] shows that if 푛 = The questions here are whether a solution 푢 of (5) exists, if 1, 2 flat measures are the only examples of 푛-uniform mea- so how regular it is, and whether there is a formula in terms sures. In 1987 Kowalski and Preiss [44] showed that if of 푓 to describe it. We say that Ω is regular if for all 푓 ∈ 푚 = 푛 + 1 and 푛 ≥ 3 then, modulo rotation and transla- 퐶(휕Ω), any solution 푢 of(5) is in 퐶(Ω) ∩ 푊1,2(Ω). In tion, Λ the support of an 푛-uniform measure 휈 is either: 푛 1923 Wiener [60] provided a remarkable characterization •Λ=ℝ ×{0} of regular domains using capacity. If Ω is regular, then for or 푥 ∈ Ω 푓 ∈ 퐶(휕Ω) 푢 ∈ 퐶(Ω) 푛+1 2 2 2 2 and if is the solution to •Λ={(푥1, 푥2, 푥3, 푥4,⋯, 푥푛+1)∈ℝ ∶푥 =푥 +푥 +푥 }. 4 1 2 3 (5), by the Maximum Principle |푢(푥)| ≤ max휕Ω |푓|. Thus See figure above. for 푥 ∈ Ω, 푇푥 ∶ 퐶(휕Ω) → ℝ defined by 푇푥(푓) = 푢(푥) Thirty years later Nimer [55] produced the first exam- is a bounded linear operator, with ‖푇푥‖ ≤ 1. Moreover ples of 푛-uniform measures in higher co- 푇푥(1) = 1. Hence, by the Riesz Representation Theorem, dimensions. His argument has an important combinato- there is a probability measure 휔푥, the harmonic measure rial component and uses Archimedes’ theorem, namely with pole at 푥, satisfying that, in ℝ3 the surface measure of the intersection of the unit sphere with a ball of small radius 푟 and centered on 푥 2 푢(푥) = ∫ 푓(푞)푑휔 (푞). (6) the sphere is exactly 휋푟 . He classifies up to isometry all 휕Ω conical 3-uniform measures in ℝ5 and produces families If Ω is regular and connected, the Harnack Principle im- of examples in any co-dimension. 푥 푦 Much time has elapsed between the initial work of Preiss plies that for 푥, 푦 ∈ Ω, 휔 and 휔 are mutually abso- and collaborators and the next set of examples. This illus- lutely continuous. Thus we will often drop the pole depen- dence and simply refer to the harmonic measure 휔 rather trates a trend in this area. Indeed for years Preiss’s work 푥 was perceived as somewhat impenetrable. In the early-to- than 휔 . mid 2000s several successful attempts were made to under- The question of whether the behavior of the harmonic stand and apply some segments of the paper (see [25, 39, measure on a given domain yields information about the 46, 57]). De Lellis [28] produced a more comprehensive structure of the boundary of the domain has attracted con- version of a special case of the argument, which has made siderable interest over the last century, with a period of this work more accessible. intense activity over the last two decades. The initial results in ℝ2 are very satisfactory. For a simply-connected

476 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 4 domain Ω ⊂ ℝ2, bounded by a Jordan curve, the bound- Lewis, Verchota, and Vogel reexamined Wolff’s construc- 푛+1 ary is a disjoint union, with the following properties: tion and were able to produce “Wolff snowflakes” in ℝ , 푛 ≥ 2 for which either ℋ − dim 휔± < 푛 or ℋ − 휕Ω = 퐺 ∪ 푆 ∪ 푁 (7) ± ± dim 휔 > 푛, where 휔 denote the harmonic measure 1. In 퐺, 휔, and ℋ1 are mutually absolutely continuous, of Ω±. They also observed, as a consequence of the mono- + − + which we denote by 휔 ≪ ℋ1 ≪ 휔, tonicity formula in [2], that if 휔 ≪ 휔 ≪ 휔 , then ± 2. Every point of 퐺 is the vertex of a cone in Ω. More- ℋ − dim 휔 ≥ 푛. 2 over if 퐶 denotes the set of “cone points” of 휕Ω, then Returning to the case of 푛 = 1, when Ω ⊂ ℝ is again ℋ1(퐶\퐺) = 0 and 휔(퐶\퐺) = 0. simply connected, and bounded by a Jordan curve, Bishop, 3. 휔(푁) = 0 and ℋ1(푆) = 0. Carleson, Garnett, and Jones [19] showed that if 퐸 ⊂ 휕Ω, ± + − 4. 푆 consists (휔 a.e.) of “twist points” (see [31] for the 휔 (퐸) > 0, then 휔 and 휔 are mutually singular (i.e. definition). 휔+ ⟂ 휔−) on 퐸 if and only if ℋ1(푇푛(휕Ω) ∩ 퐸) = 0, 5. For 휔 a.e. 푞 ∈ 퐺, the 1-density of 휔 exists and where 푇푛(휕Ω) ⊂ 휕Ω is the set of points in 휕Ω where 휃1(휔, 푞) ∈ (0, ∞) (see (3)). 휕Ω has a unique tangent line. Let 퐸 ⊂ 휕Ω be such that + − + ± 6. At 휔 a.e. point 푞 ∈ 푆 we have 휔 ≪ 휔 ≪ 휔 on 퐸 and 휔 (퐸) > 0. Then, because 휔± 0 퐸 ⊂ 푇푛(휕Ω) 휔(퐵(푞, 푟)) of [19], modulo sets of measure , . Us- lim sup = +∞, ing Beurling’s inequality, i.e., the fact that for 푞 ∈ 휕Ω and 푟 푟→0 푟 > 0, 휔+(퐵(푞, 푟))휔−(퐵(푞, 푟)) ≤ 퐶푟2, and the charac- 휔(퐵(푞, 푟)) ± ± ± lim inf = 0. terization above where 휕Ω = 퐺 ∪푆 ∪푁 (see (7)), we 푟→0 푟 conclude that 휔+ ≪ ℋ1 ≪ 휔− ≪ 휔+ on 퐸. Thus, the These results are a combination of work of Makarov, set of mutual absolute continuity of 휔−, 휔+ is a subset McMillan, Pommerenke, and Choi. See [31] for the pre- of 퐺+ ∩ 퐺− and hence of Hausdorff dimension 1. cise references. In [18], motivated by this last result, Bishop asked if Recall that the Hausdorff dimension of 휔 (denoted by in the case of ℝ푛+1, 푛 ≥ 2, the fact that 휔−, 휔+ are ℋ − dim 휔) is defined by mutually absolutely continuous on a set 퐸 ⊂ 휕Ω, with 휔±(퐸) > 0, implies that 휔± are mutually absolutely con- ℋ − dim 휔 = inf {푘 ∶ there exists 퐸 ⊂ 휕Ω (8) 푛 tinuous with respect to ℋ on 퐸 and hence dimℋ(퐸) = 푘 with ℋ (퐸) = 0 and 푛, where dimℋ denotes the Hausdorff dimension of a set. 휔(퐸 ∩ 퐾) = 휔(휕Ω ∩ 퐾) Two Phase Case 푛+1 for all compact sets 퐾 ⊂ ℝ } While in [40] and [41] we had already used some of the Important work of Makarov [47] shows that for simply properties of the tangent measures, Bishop’s question plus connected domains in ℝ2, ℋ − dim 휔 = 1, establishing the desire to understand the Wolff snowflakes better led Oksendal’s conjecture (i.e. for what type of domains in us to dig deep into Preiss’s work [56]. There we found the 푛+1 necessary tools to start tackling the problem of describing ℝ is ℋ − dim 휔 = 푛?) in dimension 2. Carleson ± [22], and Jones and Wolff [38] proved that, in general, the boundary in terms of 휔 . In [39, Kenig, Preiss and for domains in ℝ2 with a well defined harmonic measure the author] prove the following result: 휔, ℋ − dim 휔 ≤ 1. Bourgain showed that there exists 푛 ≥ 3 Ω ⊂ ℝ푛+1 푛+1 Theorem 1 ([39]). For , if is a 2-sided 휖(푛) > 0 such that for domains in ℝ with a well de- NTA domain, then fined harmonic measure 휔, ℋ−dim 휔 ≤ 푛+1−휖(푛), see [21]. Finding the optimal bound for this Hausdorff 휕Ω = Γ ∪ 푆 ∪ 푁, (9) dimension is an important open question in the area. 1. 휔+ ⟂ 휔− on 푆 and 휔±(푁) = 0. T. Wolff [61] showed, by a deep example, that, for 푛 ≥ + − + 2. On Γ, 휔 ≪ 휔 ≪ 휔 , dimℋ Γ ≤ 푛. 2, Oksendal’s conjecture (that ℋ − dim 휔 = 푛) fails. ± 3. If 휔 (Γ) > 0, dimℋ Γ = 푛. He constructed what are known as “Wolff snowflakes,” do- 푛 3 4. If ℋ 휕Ω is a Radon measure then Γ is 푛-rectifiable, ℝ ℋ − dim 휔 > 2 − 푛 + − mains in , for which and others for and 휔 ≪ ℋ ≪ 휔 ≪ 휔 on Γ. which ℋ − dim 휔 < 2. In Wolff’s construction, the domains have a certain weak regularity property. They are As a consequence there can be no Wolff snowflake for non-tangentially accessible domains (NTA) in the sense of which 휔+, 휔− are mutually absolutely continuous. The- [37]. In fact, they are 2-sided NTA domains (i.e. Ω+ = Ω orem 1 answered Bishop’s question under the assumption and Ω− = int(Ω푐) are both NTA) which plays an impor- that ℋ푛 휕Ω is a Radon measure. The general case was tant role in his estimates. NTA domains are open, con- left open, and it was clear that a new idea was needed to nected, and Wiener regular in a quantitative way. In [45], deal with the main obstacle, namely the set of points for

APRIL 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 477 which the 푛-density of the harmonic measure is 0. A note- Theorem 2 ([8]). For 푛 ≥ 2, if Ω ⊂ ℝ푛+1 is a 2-sided NTA worthy issue in this branch of GMT is that difficulties often domain, then arise from either a measure that is not locally finite or a 휕Ω = 퐺 ∪ 푆 ∪ 푁′, (10) measure whose appropriate density is zero. The proof of Theorem 1 uses tools from the theory of 1. 휔+ ⟂ 휔− on 푆 and 휔±(푁′) = 0. non-tangentially accessible domains (NTA) introduced by 2. 퐺 is 푛-rectifiable and 휔− ≪ ℋ푛 ≪ 휔+ ≪ 휔−. Jerison and Kenig [37], the monotonicity formula of Alt, Caffarelli, and Friedman [2], the theory of tangent mea- The main innovation in [8] is the introduction of a new 푛 sures introduced by Preiss [56], and the blow up techniques set of ideas involving the -dimensional Riesz transform. for harmonic measures at infinity for unbounded NTA do- In particular they use a result by Girela-Sarrión and Tolsa mains due to Kenig and Toro [40, 41]. For additional re- [32] concerning the connection between Riesz transforms sults along these lines see [11–14,29]. and quantititative rectifiability for general Radon measures. We describe the main steps to emphasize the similari- This allows them to deal with the set of points for which 푛 ties with the train of thought present in Preiss’s work. To the -density of the harmonic measure is 0. The connec- accomplish our objective, we use the blow-up analysis de- tion between the Riesz transform and harmonic measure veloped in [41]. At 휔± a.e. point on the set where 휔+ stems from the fact that the Riesz kernel is the gradient and 휔− are mutually absolutely continuous, the tangent of the Newtonian potential. The relationship between the measures to 휔± (in the sense of [50], [56]) are harmonic Riesz transform and rectifiability has been an important measures associated to the Laplacian on domains where a component in the development of quantitative geometric harmonic polynomial is either positive or negative. The re- measure theory, a field initiated by David and Semmes (see sulting harmonic measure is supported on the zero set of [26],[27]) in the early 1990s, and embraced by a large com- this harmonic polynomial. Using the fact that for almost munity. Quantitative GMT has developed into a vibrant every point a tangent measure to a tangent measure is a area in which several important milestones have been ac- tangent measure (see [50]) and the fact that the zero set complished in recent years (e.g. the solution of the David- 푛+1 Semmes conjecture by Nazarov, Tolsa, and Volberg [53, of a harmonic polynomial in ℝ is smooth except for a 54]). In a subsequent paper, Azzam, Mourgoglou, Tolsa, set of Hausdorff dimension 푛 − 1 (see [33]), one shows ± and Volberg significantly relax the hypothesis on the do- that at 휔 a.e. point on this set, 푛-flat measures always ± mains for which Theorem 2 holds [10]. arise as tangent measures to 휔 . They correspond to lin- We note that the narrative started with a question from ear harmonic polynomials. We then show and this is the potential analysis. The initial results were the product of crucial step, that if one tangent measure is flat on the set of a successful approach taking a GMT point of view. Once mutual absolute continuity, then all tangent measures are this work was in place, questions arose that required deep flat. To accomplish this we use a connectivity argument results in harmonic analysis and quantitative GMT to be similar to the one from [56]. The key point is that if a tackled. The final outcome lies in the interface of poten- tangent measure is not flat, being the harmonic measure tial theory and geometric measure theory. These results supported on the zero set of a harmonic polynomial of illustrate how the synergy between very distinct areas can degree higher than 1, its tangent measure at infinity is far produce truly unique and unexpected results. from flat, and a connectivity argument, as in [56], gives ± a contradiction. Modulo a set of 휔 measure 0, Γ as in One Phase Case (10), is the set of mutual absolute continuity for which one In the Harmonic Measure section, we started by asking (and hence all) tangent measures are 푛-flat. An easy argu- whether the behavior of the harmonic measure of a do- ment then shows that dim Γ ≤ 푛. To conclude that if ℋ main yields information about the structure of its bound- 휔±(Γ) > 0, dim Γ = 푛, one uses the Alt-Caffarelli- ℋ ary (one phase case). The discussion very quickly turned Friedman monotonicity formula of [2] as in [45]. This to the situation where we consider the harmonic measures yields a version of Beurling’s inequality in higher dimen- of a set and its complement (two phase case). The ratio- sions. If 휎 = ℋ푛 휕Ω, the surface measure to the bound- nale was that the two phase case was more clearly related ary, is a Radon measure, we show that the 푛-density of to GMT. We now return to the one phase case, where both 휎 is 1 a.e., which by Preiss’s theorem ensures that Γ is 푛- the quantitative and qualitative questions have sparked an rectifiable. incredible amount of interest, generating lots of activity In a remarkable paper, Azzam, Mourgoglou, and Tolsa that has culminated in truly optimal results. [8] answer Bishop’s question completely. Although their In 1916 F. and M. Riesz proved that for a simply con- result holds in greater generality, we state it here in the nected domain in the complex plane with a rectifiable context of the discussion above for simplicity. boundary, harmonic measure is absolutely continuous with respect to arc length measure on the boundary [58].

478 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 4 Bishop and Jones [20] have shown that in this type of do- 54] where the authors prove the David-Semmes conjecture main, if only a portion of the boundary is rectifiable, then in co-dimension 1; that is, they show that the bounded- harmonic measure is absolutely continuous with respect ness of the Riesz transform of a measure implies its rectifi- to arc length on that portion. They also showed that the ability. result of [58] may fail in the absence of some topological Note that the description above of the results in the hypothesis (e.g., simple connectedness). Examples con- area does not include a qualitative version of the F. and M. structed in [62] and [63] show that, in higher dimensions, Riesz type result in higher dimensions. At this stage, works some topological restrictions, even stronger than those nee- of [1, 9, 15] indicate that obtaining an optimal condition ded in the planar case, are required for the absolute conti- on a domain to ensure that rectifiability of the boundary nuity of 휔 with respect to surface measure to the bound- implies absolute continuity of harmonic measure with re- ary. spect to the surface measure is challenging. Higher dimensional analogues of this question have This field has been evolving in several interesting new played a central role in the development of the study of directions. They all fit under the umbrella of understand- partial differential equations in non-smooth domains. In ing the structure of the support of a measure associated to 1982 Dahlberg [23] showed the harmonic measure of a a differential operator in a canonical way. One direction Lipschitz domain and the surface measure to its bound- concerns understanding questions similar to those discuss- ary are mutually absolutely continuous (in a quantitative ed in the three previous sections for the elliptic measure scale-invariant way, namely 휔 ∈ 퐴∞(휎)). Similar results of a uniformly elliptic second order divergence form op- hold on chord arc domains (these are NTA domains for erator. Another one looks at the problems analogous to which the surface measure to the boundary is Ahlfors reg- those appearing in the Harmonic Measure and One Phase ular; that is, the surface measure of a ball centered on the Case sections for the elliptic measure corresponding to a boundary and of radius 푟 grows like 푟푛) (see [24], [59]). degenerate elliptic operator on a domain in ℝ푛+1 whose The relationship between quantitative absolute continu- boundary has dimension strictly less than 푛. The unifying ity properties of harmonic measure with respect to surface trait is the beautiful cross-pollination between geometric measure and the regularity of the boundary (also expressed measure theory, potential theory, harmonic analysis, and in quantitative terms) is now very well understood, see for partial differential equation. The expectation is that this example [3, 6, 7, 15, 34–36]. synergy will continue to uncover unsuspected connections, Here we only focus on the optimal qualitative result leading to the development of the field in ways that are that provides a complete answer to Bishop’s question. In a only possible thanks to the contributions from a diverse very interesting piece of work, Azzam, Hofmann, Martell, group of analysts. Mayboroda, Mourgoglou, Tolsa, and Volberg show a converse to the results in [20,58] in all dimensions. See [4,5] References [1] Akman M, Badger M, Hofmann S, Martell JM. Rectifia- 푛+1 Theorem 3 ([4], [5]). Let Ω ⊂ ℝ be an open connected bility and elliptic measures on 1-sided NTA domains with set and let 휔 be the harmonic measure in Ω. Let 퐸 ⊂ 휕Ω Ahlfors-David regular boundaries, Trans. Amer. Math. Soc., with Hausdorff measure ℋ푛(퐸) < ∞. (369, no. 8): 5711–5745, 2017. MR3646776 • If 휔 is absolutely continuous with respect to ℋ푛 on [2] Alt HW, Caffarelli LA, Friedman A. Variational problems with two phases and their free boundaries, Trans. Amer. 퐸, then 휔 restricted to 퐸 is an 푛-rectifiable measure. Math. Soc., (282, no. 2): 431–461, 1984. MR732100 • If ℋ푛 is absolutely continuous with respect to 휔 on [3] Azzam J. Semi-uniform domains and the 퐴∞ property for 퐸, then 퐸 is an 푛-recitfiable set. harmonic measure, arXiv:1711.03088. This theorem can be understood as a free boundary reg- [4] Azzam J, Hofmann S, Martell JM, Mayboroda S, Mour- ularity problem for the harmonic measure (an initial exam- goglou M. 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Harmonic measure and the corresponding Green function that allow them to and quantitative connectivity: geometric characterization show that the Riesz transform is a bounded operator. Then of the 퐿푝 solvability of the dirichlet problem. Part II. they appeal the work of Nazarov, Tolsa, and Volberg [53, arXiv:1903.07975.

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