Rectifiability of harmonic measure

Jonas Azzam; Steve Hofmann; José María Martell; Svitlana Mayboroda; Mihalis Mourgoglou; Xavier Tolsa; Alexander Volberg

doi:10.1007/s00039-016-0371-x

Rectifiability of harmonic measure

Jonas Azzam, Steve Hofmann, José María Martell, Svitlana Mayboroda, Mihalis Mourgoglou, Xavier Tolsa, Alexander Volberg

School of Mathematics

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

53 Scopus citations

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 (US)
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
State	Published - Jun 1 2016

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© 2016, Springer International Publishing.

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AU - Azzam, Jonas

AU - Hofmann, Steve

AU - Martell, José María

AU - Mayboroda, Svitlana

AU - Mourgoglou, Mihalis

AU - Tolsa, Xavier

AU - Volberg, Alexander

PY - 2016/6/1

Y1 - 2016/6/1

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.

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Rectifiability of harmonic measure

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