Rectifiability of harmonic measure in domains with porous boundaries

Abstract

We show that if $n\ge 1$, $\Omega \subset {\mathbb{R}}^{n+1}$ is a connected domain with porous boundary, and $E\subset \partial \Omega$ is a set of finite and positive Hausdorff $H^n$-measure upon which the harmonic measure $\omega$ is absolutely continuous with respect to $H^n$, then $\omega {|}_E$ is concentrated on an $n$-rectifiable set.

arXiv commentIn this version the main theorem is also proved in the planar case

References in corpus (2)

• Null sets of harmonic measure on NTA domains: Lipschitz approximation revisited
• The $Tb$-theorem on non-homogeneous spaces that proves a conjecture of Vitushkin

Cited by in corpus (4)

• Rectifiability of harmonic measure
• Rectifiability and elliptic measures on 1-sided NTA domains with Ahlfors-David regular boundaries
• Uniform domains with rectifiable boundaries and harmonic measure
• Absolute continuity between the surface measure and harmonic measure implies rectifiability