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Xavier Tolsa - One of the best experts on this subject based on the ideXlab platform.
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on a two phase problem for harmonic measure in general domains
American Journal of Mathematics, 2019Co-Authors: Jonas Azzam, Mihalis Mourgoglou, Xavier Tolsa, Alexander VolbergAbstract:We show that, for disjoint domains in the Euclidean space, mutual absolute continuity of their harmonic measures implies absolute continuity with respect to surface measure and Rectifiability in the intersection of their boundaries. This improves on our previous result which assumed that the boundaries satisfied the capacity density condition.
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Rectifiability of harmonic measure
Geometric and Functional Analysis, 2016Co-Authors: Jonas Azzam, Steve Hofmann, Mihalis Mourgoglou, Xavier Tolsa, Jose Maria Martell, Svitlana Mayboroda, Alexander VolbergAbstract:In the present paper we prove that for any open connected set \({\Omega\subset\mathbb{R}^{n+1}}\), \({n\geq 1}\), and any \({E\subset \partial \Omega}\) with \({\mathcal{H}^n(E)<\infty}\), absolute continuity of the harmonic measure \({\omega}\) with respect to the Hausdorff measure on E implies that \({\omega|_E}\) is rectifiable. This solves an open problem on harmonic measure which turns out to be an old conjecture even in the planar case \({n=1}\).
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Mutual absolute continuity of interior and exterior harmonic measure implies Rectifiability
arXiv: Classical Analysis and ODEs, 2016Co-Authors: Jonas Azzam, Mihalis Mourgoglou, Xavier TolsaAbstract:We show that, for disjoint domains in the Euclidean space whose boundaries satisfy a non-degeneracy condition, mutual absolute continuity of their harmonic measures implies absolute continuity with respect to surface measure and Rectifiability in the intersection of their boundaries.
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on the uniform Rectifiability of ad regular measures with bounded riesz transform operator the case of codimension 1
Acta Mathematica, 2014Co-Authors: Fedor Nazarov, Alexander Volberg, Xavier TolsaAbstract:We prove that if μ is a d-dimensional Ahlfors-David regular measure in \({\mathbb{R}^{d+1}}\) , then the boundedness of the d-dimensional Riesz transform in L2(μ) implies that the non-BAUP David–Semmes cells form a Carleson family. Combined with earlier results of David and Semmes, this yields the uniform Rectifiability of μ.
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analytic capacity the cauchy transform and non homogeneous calderon zygmund theory
2014Co-Authors: Xavier TolsaAbstract:Introduction.- Basic notation.- Chapter 1. Analytic capacity.- Chapter 2. Basic Calderon-Zygmund theory with non doubling measures.- Chapter 3. The Cauchy transform and Menger curvature.- Chapter 4. The capacity gamma+.- Chapter 5. A Tb theorem of Nazarov, Treil and Volberg.- Chapter 6. The comparability between gamma and gamma +, and the semiadditivity of analytic capacity.- Chapter 7. Curvature and Rectifiability.- Chapter 8. Principal values for the Cauchy transform and Rectifiability.- Chapter 9. RBMO(mu) and H1 atb(mu).- Bibliography.- Index.
Steve Hofmann - One of the best experts on this subject based on the ideXlab platform.
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quantitative absolute continuity of harmonic measure and the dirichlet problem a survey of recent progress
Acta Mathematica Sinica, 2019Co-Authors: Steve HofmannAbstract:It is a well-known folklore result that quantitative, scale invariant absolute continuity (more precisely, the weak-A∞ property) of harmonic measure with respect to surface measure, on the bound¬ary of an open set Ω ⊂ ℝn+1 with Ahlfors-David regular boundary, is equivalent to the solvability of the Dirichlet problem in Ω, with data in Lp(∂ Ω) for some p < ∞. Drawing an analogy to the famous Wiener criterion, which characterizes the domains in which the classical Dirichlet problem, with contin¬uous boundary data, can be solved, one may seek to characterize the open sets for which Lp solvability holds, thus allowing for singular boundary data. It has been known for some time that absolute continuity of harmonic measure is closely tied to Rectifiability properties of ∂ Ω, but also that Rectifiability alone is not sufficient to guarantee absolute continuity. In this note, we survey recent progress in this area, culminating in a geometric charac¬terization of the weak-A∞ property, and hence of solvability of the Lp Dirichlet problem for some finite p. This characterization, obtained under rather optimal background hypotheses, follows from a combination of the present author’s joint work with Martell, and the work of Azzam, Mourgoglou and Tolsa.
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Rectifiability of harmonic measure
Geometric and Functional Analysis, 2016Co-Authors: Jonas Azzam, Steve Hofmann, Mihalis Mourgoglou, Xavier Tolsa, Jose Maria Martell, Svitlana Mayboroda, Alexander VolbergAbstract:In the present paper we prove that for any open connected set \({\Omega\subset\mathbb{R}^{n+1}}\), \({n\geq 1}\), and any \({E\subset \partial \Omega}\) with \({\mathcal{H}^n(E)<\infty}\), absolute continuity of the harmonic measure \({\omega}\) with respect to the Hausdorff measure on E implies that \({\omega|_E}\) is rectifiable. This solves an open problem on harmonic measure which turns out to be an old conjecture even in the planar case \({n=1}\).
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uniform Rectifiability and harmonic measure iv ahlfors regularity plus poisson kernels in l p implies uniform Rectifiability
arXiv: Classical Analysis and ODEs, 2015Co-Authors: Steve Hofmann, Jose Maria MartellAbstract:Let $E\subset \mathbb{R}^{n+1}$, $n\ge 2$, be an Ahlfors-David regular set of dimension $n$. We show that the weak-$A_\infty$ property of harmonic measure, for the open set $\Omega:= \mathbb{R}^{n+1}\setminus E$, implies uniform Rectifiability of $E$.
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uniform Rectifiability and harmonic measure ii poisson kernels in lp imply uniform Rectifiability
Duke Mathematical Journal, 2014Co-Authors: Steve Hofmann, Jose Maria Martell, Ignacio UriartetueroAbstract:We present the converse to a higher-dimensional, scale-invariant version of the classical F. and M. Riesz theorem, proved by the first two authors. More precisely, for n≥2, for an Ahlfors–David regular domain Ω⊂Rn+1 which satisfies the Harnack chain condition plus an interior (but not exterior) corkscrew condition, we show that absolute continuity of the harmonic measure with respect to the surface measure on ∂Ω, with scale-invariant higher integrability of the Poisson kernel, is sufficient to imply quantitative Rectifiability of ∂Ω.
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uniform Rectifiability and harmonic measure i uniform Rectifiability implies poisson kernels in l p
Annales Scientifiques De L Ecole Normale Superieure, 2014Co-Authors: Steve Hofmann, Jose Maria MartellAbstract:The first author was supported by NSF grant DMS-0801079. The second author was supported by MINECO Grant MTM2010-16518 and ICMAT Severo Ochoa project SEV-2011-0087.
Jose Maria Martell - One of the best experts on this subject based on the ideXlab platform.
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Rectifiability of harmonic measure
Geometric and Functional Analysis, 2016Co-Authors: Jonas Azzam, Steve Hofmann, Mihalis Mourgoglou, Xavier Tolsa, Jose Maria Martell, Svitlana Mayboroda, Alexander VolbergAbstract:In the present paper we prove that for any open connected set \({\Omega\subset\mathbb{R}^{n+1}}\), \({n\geq 1}\), and any \({E\subset \partial \Omega}\) with \({\mathcal{H}^n(E)<\infty}\), absolute continuity of the harmonic measure \({\omega}\) with respect to the Hausdorff measure on E implies that \({\omega|_E}\) is rectifiable. This solves an open problem on harmonic measure which turns out to be an old conjecture even in the planar case \({n=1}\).
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uniform Rectifiability and harmonic measure iv ahlfors regularity plus poisson kernels in l p implies uniform Rectifiability
arXiv: Classical Analysis and ODEs, 2015Co-Authors: Steve Hofmann, Jose Maria MartellAbstract:Let $E\subset \mathbb{R}^{n+1}$, $n\ge 2$, be an Ahlfors-David regular set of dimension $n$. We show that the weak-$A_\infty$ property of harmonic measure, for the open set $\Omega:= \mathbb{R}^{n+1}\setminus E$, implies uniform Rectifiability of $E$.
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uniform Rectifiability and harmonic measure ii poisson kernels in lp imply uniform Rectifiability
Duke Mathematical Journal, 2014Co-Authors: Steve Hofmann, Jose Maria Martell, Ignacio UriartetueroAbstract:We present the converse to a higher-dimensional, scale-invariant version of the classical F. and M. Riesz theorem, proved by the first two authors. More precisely, for n≥2, for an Ahlfors–David regular domain Ω⊂Rn+1 which satisfies the Harnack chain condition plus an interior (but not exterior) corkscrew condition, we show that absolute continuity of the harmonic measure with respect to the surface measure on ∂Ω, with scale-invariant higher integrability of the Poisson kernel, is sufficient to imply quantitative Rectifiability of ∂Ω.
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uniform Rectifiability and harmonic measure i uniform Rectifiability implies poisson kernels in l p
Annales Scientifiques De L Ecole Normale Superieure, 2014Co-Authors: Steve Hofmann, Jose Maria MartellAbstract:The first author was supported by NSF grant DMS-0801079. The second author was supported by MINECO Grant MTM2010-16518 and ICMAT Severo Ochoa project SEV-2011-0087.
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uniform Rectifiability and harmonic measure iii riesz transform bounds imply uniform Rectifiability of boundaries of 1 sided nta domains
International Mathematics Research Notices, 2014Co-Authors: Steve Hofmann, Jose Maria Martell, Svitlana MayborodaAbstract:Let E R n+1 , n 2, be a closed, Ahlfors-David regular set of di- mension n satisfying the "Riesz Transform bound" sup ">0 Z E Z fy2E:jx yj>"g x y
Martell, José María - One of the best experts on this subject based on the ideXlab platform.
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Uniform Rectifiability and harmonic measure, II: Poisson kernels in Lp imply uniform Rectifiability
'Duke University Press', 2020Co-Authors: Hofman Steve, Martell, José María, Uriarte-tuero IgnacioAbstract:We present the converse to a higher-dimensional, scale-invariant version of the classical F. and M. Riesz theorem, proved by the first two authors. More precisely, for n ≥ 2, for an Ahlfors-David regular domain Ω ⊂ ℝn+1 which satisfies the Harnack chain condition plus an interior (but not exterior) corkscrew condition, we show that absolute continuity of the harmonic measure with respect to the surface measure on ∂Ω, with scale-invariant higher integrability of the Poisson kernel, is sufficient to imply quantitative Rectifiability of ∂Ω.The first author was supported by NSF grants DMS-0801079 and DMS-1101244. The second author was supported by MICINN Grant MTM2010-16518, and by CSIC PIE 200850I015. The third author was partially supported by grants DMS-0901524, CAREER DMS- 1056965 (US NSF), Sloan Research Foundation, and MTM2010-16232, MTM2009-14694-C02-01 (Spain).Peer Reviewe
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Rectifiability of harmonic measure
'Springer Science and Business Media LLC', 2020Co-Authors: Azzam, Jonas, Hofmann Steve, Martell, José María, Mayboroda, Svitlana, Mourgoglou Mihalis, Tolsa Xavier, Volberg AlexanderAbstract:In the present paper we prove that for any open connected set Ω ⊂ R, n≥ 1 , and any E⊂ ∂Ω with H(E) < ∞, absolute continuity of the harmonic measure ω with respect to the Hausdorff measure on E implies that ω| is rectifiable. This solves an open problem on harmonic measure which turns out to be an old conjecture even in the planar case n= 1.The second author was supported in part by NSF grant DMS 1361701. The third author has been partially supported by ICMAT Severo Ochoa project SEV-2011- 0087 and he acknowledges that the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC agreement no. 615112 HAPDEGMT. The fourth author is supported in part by the Alfred P. Sloan Fellowship, the NSF INSPIRE Award DMS 1344235, NSF CAREER Award DMS 1220089 and NSF UMN MRSEC Seed grant DMR 0212302. The sixth author was supported by the ERC grant 320501 of the European Research Council (FP7/2007-2013) (which also funded the first and fifth authors), by 2014- SGR-75 (Catalonia), MTM2013-44304-P (Spain), and by the Marie Curie ITN MAnET (FP7-607647). The last author was partially supported by the NSF grant DMS-126554
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Uniform Rectifiability and harmonic measure i: uniform Rectifiability implies Poisson kernels in Lp
'Societe Mathematique de France', 2020Co-Authors: Hofmann Steve, Martell, José MaríaAbstract:We present a higher dimensional, scale-invariant version of a classical theorem of F. and M. Riesz [37]. More precisely, we establish scale invariant absolute continuity of harmonic measure with respect to surface measure, along with higher integrability of the Poisson kernel, for a domain Ω ⊂ Rn+1; n 2, with a uniformly rectiable boundary, which satises the Harnack chain condition plus an interior (but not exterior) Corkscrew condition. In a companion paper to this one [28], we also establish a converse, in which we deduce uniform Rectifiability of the boundary, assuming scale invariant Lq bounds, with q > 1, on the Poisson kernel.The first author was supported by NSF grant DMS-0801079. The second author was supported by MINECO Grant MTM2010-16518 and ICMAT Severo Ochoa project SEV-2011-0087.Peer Reviewe
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Rectifiability, interior approximation and Harmonic measure
'International Press of Boston', 2019Co-Authors: Akman Murat, Bortz Simon, Hofmann Steve, Martell, José MaríaAbstract:We prove a structure theorem for any $n$-rectifiable set $E\subset\mathbb{R}^{n+1}, n \geq 1$, satisfying a weak version of the lower ADR condition, and having locally finite $\mathcal{H}^{n}$ ($n$-dimensional Hausdorff) measure. Namely, that $\mathcal{H}^{n}$-almost all of $E$ can be covered by a countable union of boundaries of bounded Lipschitz domains contained in $\mathbb{R}^{n+1}\setminus E$. As a consequence, for harmonic measure in the complement of such a set $E$, we establish a non-degeneracy condition which amounts to saying that $\mathcal{H}^{n}|_{E}$ is ''absolutely continuous'' with respect to harmonic measure in the sense that any Borel subset of $E$ with strictly positive $\mathcal{H}^{n}$ measure has strictly positive harmonic measure in some connected component of $\mathbb{R}^{n+1}\setminus E$. We also provide some counterexamples showing that our result for harmonic measure is optimal. Moreover, we show that if, in addition, a set $E$ as above is the boundary of a connected domain $\Omega\subset\mathbb{R}^{n+1}$ which satisfies an infinitesimal interior thickness condition, then $\mathcal{H}^{n}|_{\partial\Omega}$ is absolutely continuous (in the usual sense) with respect to harmonic measure for $\Omega$. Local versions of these results are also proved: if just some piece of the boundary is $n$-rectifiable then we get the corresponding absolute continuity on that piece. As a consequence of this and recent results in [AHM$^{3}$TV], we can decompose the boundary of any open connected set satisfying the previous conditions in two disjoint pieces: one that is $n$-rectifiable where Hausdorff measure is absolutely continuous with respect to harmonic measure and another purely $n$-unrectifiable piece having vanishing harmonic measure. [AHM$^{3}$TV] J. Azzam, S. Hofmann, J.M. Martell, S. Mayboroda, M. Mourgoglou, X. Tolsa and A. Volberg. Rectifiability of harmonic measure. arXiv:1509.06294, To appear in GAFA
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Rectifiability, interior approximation and harmonic measure
'International Press of Boston', 2019Co-Authors: Akman M., Bortz S., Hofmann S., Martell, José MaríaAbstract:We prove a structure theorem for any n-rectifiable set E⊂R, n≥1, satisfying a weak version of the lower ADR condition, and having locally finite H (n-dimensional Hausdorff) measure. Namely, that H-almost all of E can be covered by a countable union of boundaries of bounded Lipschitz domains contained in R \E. As a consequence, for harmonic measure in the complement of such a set E, we establish a non-degeneracy condition which amounts to saying that H| is “absolutely continuous” with respect to harmonic measure in the sense that any Borel subset of E with strictly positive H measure has strictly positive harmonic measure in some connected component of R \E. We also provide some counterexamples showing that our result for harmonic measure is optimal. Moreover, we show that if, in addition, a set E as above is the boundary of a connected domain Ω⊂R which satisfies an infinitesimal interior thickness condition, then H| is absolutely continuous (in the usual sense) with respect to harmonic measure for Ω. Local versions of these results are also proved: if just some piece of the boundary is n-rectifiable then we get the corresponding absolute continuity on that piece. As a consequence of this and recent results in [AHMTV], we can decompose the boundary of any open connected set satisfying the previous conditions in two disjoint pieces: one that is n-rectifiable where Hausdorff measure is absolutely continuous with respect to harmonic measure and another purely n-unrectifiable piece having vanishing harmonic measure.The first and last authors acknowledge financial support from the Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa” Programme for Centres of Excellence in R&D (SEV-2015-0554). They also acknowledge that the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC agreement no. 615112 HAPDEGMT. The second and third authors were supported by NSF grant DMS-1361701.Peer Reviewe
Hofmann Steve - One of the best experts on this subject based on the ideXlab platform.
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Uniform Rectifiability and harmonic measure i: uniform Rectifiability implies Poisson kernels in Lp
'Societe Mathematique de France', 2020Co-Authors: Hofmann Steve, Martell, José MaríaAbstract:We present a higher dimensional, scale-invariant version of a classical theorem of F. and M. Riesz [37]. More precisely, we establish scale invariant absolute continuity of harmonic measure with respect to surface measure, along with higher integrability of the Poisson kernel, for a domain Ω ⊂ Rn+1; n 2, with a uniformly rectiable boundary, which satises the Harnack chain condition plus an interior (but not exterior) Corkscrew condition. In a companion paper to this one [28], we also establish a converse, in which we deduce uniform Rectifiability of the boundary, assuming scale invariant Lq bounds, with q > 1, on the Poisson kernel.The first author was supported by NSF grant DMS-0801079. The second author was supported by MINECO Grant MTM2010-16518 and ICMAT Severo Ochoa project SEV-2011-0087.Peer Reviewe
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Uniform Rectifiability implies Varopoulos extensions
2020Co-Authors: Hofmann Steve, Tapiola OlliAbstract:We construct extensions of Varopolous type for functions $f \in \text{BMO}(E)$, for any uniformly rectifiable set $E$ of codimension one. More precisely, let $\Omega \subset \mathbb{R}^{n+1}$ be an open set satisfying the corkscrew condition, with an $n$-dimensional uniformly rectifiable boundary $\partial \Omega$, and let $\sigma := \mathcal{H}^n\lfloor_{\partial \Omega}$ denote the surface measure on $\partial \Omega$. We show that if $f \in \text{BMO}(\partial \Omega,d\sigma)$ with compact support on $\partial \Omega$, then there exists a smooth function $V$ in $\Omega$ such that $|\nabla V(Y)| \, dY$ is a Carleson measure with Carleson norm controlled by the BMO norm of $f$, and such that $V$ converges in some non-tangential sense to $f$ almost everywhere with respect to $\sigma$. Our results should be compared to recent geometric characterizations of $L^p$-solvability and of BMO-solvability of the Dirichlet problem, by Azzam, the first author, Martell, Mourgoglou and Tolsa and by the first author and Le, respectively. In combination, this latter pair of results shows that one can construct, for all $f \in C_c(\partial \Omega)$, a harmonic extension $u$, with $|\nabla u(Y)|^2 \text{dist}(Y,\partial \Omega) \, dY $ a Carleson measure controlled by the BMO norm of $f$, only in the presence of an appropriate quantitative connectivity condition.Comment: 47 page
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Uniform Rectifiability and ε-approximability of harmonic functions in Lp
'Cellule MathDoc CEDRAM', 2020Co-Authors: Hofmann Steve, Tapiola OlliAbstract:Suppose that E⊂Rn+1 is a uniformly rectifiable set of codimension 1. We show that every harmonic function is ε-approximable in Lp(Ω) for every p∈(1,∞), where Ω:=Rn+1∖E. Together with results of many authors this shows that pointwise, L∞ and Lp type ε-approximability properties of harmonic functions are all equivalent and they characterize uniform Rectifiability for codimension 1 Ahlfors–David regular sets. Our results and techniques are generalizations of recent works of T. Hytönen and A. Rosén and the first author, J. M. Martell and S. Mayboroda.peerReviewe
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Rectifiability, interior approximation and Harmonic measure
'International Press of Boston', 2019Co-Authors: Akman Murat, Bortz Simon, Hofmann Steve, Martell, José MaríaAbstract:We prove a structure theorem for any $n$-rectifiable set $E\subset\mathbb{R}^{n+1}, n \geq 1$, satisfying a weak version of the lower ADR condition, and having locally finite $\mathcal{H}^{n}$ ($n$-dimensional Hausdorff) measure. Namely, that $\mathcal{H}^{n}$-almost all of $E$ can be covered by a countable union of boundaries of bounded Lipschitz domains contained in $\mathbb{R}^{n+1}\setminus E$. As a consequence, for harmonic measure in the complement of such a set $E$, we establish a non-degeneracy condition which amounts to saying that $\mathcal{H}^{n}|_{E}$ is ''absolutely continuous'' with respect to harmonic measure in the sense that any Borel subset of $E$ with strictly positive $\mathcal{H}^{n}$ measure has strictly positive harmonic measure in some connected component of $\mathbb{R}^{n+1}\setminus E$. We also provide some counterexamples showing that our result for harmonic measure is optimal. Moreover, we show that if, in addition, a set $E$ as above is the boundary of a connected domain $\Omega\subset\mathbb{R}^{n+1}$ which satisfies an infinitesimal interior thickness condition, then $\mathcal{H}^{n}|_{\partial\Omega}$ is absolutely continuous (in the usual sense) with respect to harmonic measure for $\Omega$. Local versions of these results are also proved: if just some piece of the boundary is $n$-rectifiable then we get the corresponding absolute continuity on that piece. As a consequence of this and recent results in [AHM$^{3}$TV], we can decompose the boundary of any open connected set satisfying the previous conditions in two disjoint pieces: one that is $n$-rectifiable where Hausdorff measure is absolutely continuous with respect to harmonic measure and another purely $n$-unrectifiable piece having vanishing harmonic measure. [AHM$^{3}$TV] J. Azzam, S. Hofmann, J.M. Martell, S. Mayboroda, M. Mourgoglou, X. Tolsa and A. Volberg. Rectifiability of harmonic measure. arXiv:1509.06294, To appear in GAFA
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Rectifiability, interior approximation and Harmonic Measure
'International Press of Boston', 2018Co-Authors: Akman Murat, Bortz Simon, Hofmann Steve, Martell, José MaríaAbstract:We prove a structure theorem for any $n$-rectifiable set $E\subset \mathbb{R}^{n+1}$, $n\ge 1$, satisfying a weak version of the lower ADR condition, and having locally finite $H^n$ ($n$-dimensional Hausdorff) measure. Namely, that $H^n$-almost all of $E$ can be covered by a countable union of boundaries of bounded Lipschitz domains contained in $\mathbb{R}^{n+1}\setminus E$. As a consequence, for harmonic measure in the complement of such a set $E$, we establish a non-degeneracy condition which amounts to saying that $H^n|_E$ is "absolutely continuous" with respect to harmonic measure in the sense that any Borel subset of $E$ with strictly positive $H^n$ measure has strictly positive harmonic measure in some connected component of $\mathbb{R}^{n+1}\setminus E$. We also provide some counterexamples showing that our result for harmonic measure is optimal. Moreover, we show that if, in addition, a set $E$ as above is the boundary of a connected domain $\Omega \subset \mathbb{R}^{n+1}$ which satisfies an infinitesimal interior thickness condition, then $H^n|_{\partial\Omega}$ is absolutely continuous (in the usual sense) with respect to harmonic measure for $\Omega$. Local versions of these results are also proved: if just some piece of the boundary is $n$-rectifiable then we get the corresponding absolute continuity on that piece. As a consequence of this and recent results by Azzam-Hofmann-Martell-Mayboroda-Mourgoglou-Tolsa-Volberg, we can decompose the boundary of any open connected set satisfying the previous conditions in two disjoint pieces: one that is $n$-rectifiable where Hausdorff measure is absolutely continuous with respect to harmonic measure and another purely $n$-unrectifiable piece having vanishing harmonic measure