Absolute continuity between the surface measure and harmonic measure implies rectifiability

Abstract

In the present paper we prove 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.

References in corpus (2)

• Uniform Rectifiability and harmonic measure IV: Ahlfors regularity plus Poisson kernels in $L^p$ implies uniform rectifiability
• Rectifiability of harmonic measure in domains with porous boundaries