Harmonic measure is rectifiable if it is absolutely continuous with respect to the co-dimension one Hausdorff measure

Abstract

In the present paper we sketch the proof of the fact that for any open connected set $\Omega \subset {\mathbb{R}}^{n+1}$, $n\ge 1$, and any $E\subset \partial \Omega$ with $0<{\mathcal{H}}^n(E)<\infty$, absolute continuity of the harmonic measure $\omega$ with respect to the Hausdorff measure on $E$ implies that $\omega {|}_E$ is rectifiable.

arXiv admin note: text overlap with arXiv:1509.06294

References in corpus (4)

• Rectifiability of harmonic measure
• Uniform Rectifiability and harmonic measure IV: Ahlfors regularity plus Poisson kernels in $L^p$ implies uniform rectifiability
• The $Tb$-theorem on non-homogeneous spaces that proves a conjecture of Vitushkin
• Absolute continuity between the surface measure and harmonic measure implies rectifiability