Rectifiability of harmonic measure

Jonas Azzam; Steve Hofmann; Jose Maria Martell; Svitlana Mayboroda; Mihalis Mourgoglou; Xavier Tolsa; Alexander Volberg

doi:10.1007/s00039-016-0371-x

Rectifiability of harmonic measure

Jonas Azzam
, Steve Hofmann
, Jose Maria Martell
, Svitlana Mayboroda
, Mihalis Mourgoglou
, Xavier Tolsa
, Alexander Volberg

Research output: Contribution to journal › Article › peer-review

Abstract

In the present paper we prove that for any open connected set Ω⊂Rn+1, n≥1, and any E⊂∂Ω with Hn(E)<∞, absolute continuity of the harmonic measure ω with respect to the Hausdorff measure on E implies that ω|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.

Original language	English
Pages (from-to)	703-728
Number of pages	26
Journal	Geometric and Functional Analysis
Volume	26
Issue number	3
DOIs	https://doi.org/10.1007/s00039-016-0371-x
Publication status	Published - 16 Jun 2016
Externally published	Yes

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10.1007/s00039-016-0371-x

https://arxiv.org/abs/1509.06294

Cite this

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title = "Rectifiability of harmonic measure",

abstract = "In the present paper we prove that for any open connected set Ω⊂Rn+1, n≥1, and any E⊂∂Ω with Hn(E)<∞, absolute continuity of the harmonic measure ω with respect to the Hausdorff measure on E implies that ω|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.",

author = "Jonas Azzam and Steve Hofmann and Martell, \{Jose Maria\} and Svitlana Mayboroda and Mihalis Mourgoglou and Xavier Tolsa and Alexander Volberg",

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AU - Azzam, Jonas

AU - Hofmann, Steve

AU - Martell, Jose Maria

AU - Mayboroda, Svitlana

AU - Mourgoglou, Mihalis

AU - Tolsa, Xavier

AU - Volberg, Alexander

PY - 2016/6/16

Y1 - 2016/6/16

N2 - In the present paper we prove that for any open connected set Ω⊂Rn+1, n≥1, and any E⊂∂Ω with Hn(E)<∞, absolute continuity of the harmonic measure ω with respect to the Hausdorff measure on E implies that ω|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.

AB - In the present paper we prove that for any open connected set Ω⊂Rn+1, n≥1, and any E⊂∂Ω with Hn(E)<∞, absolute continuity of the harmonic measure ω with respect to the Hausdorff measure on E implies that ω|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.

U2 - 10.1007/s00039-016-0371-x

DO - 10.1007/s00039-016-0371-x

M3 - Article

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VL - 26

SP - 703

EP - 728

JO - Geometric and Functional Analysis

JF - Geometric and Functional Analysis

IS - 3

ER -

Rectifiability of harmonic measure

Abstract

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