Absolute continuity of harmonic measure for domains with lower regular boundaries

Abstract

We study absolute continuity of harmonic measure with respect to surface measure on domains $\Omega$ that have large complements. We show that if $\Gamma \subset {\mathbb{R}}^{d+1}$ is $d$-Ahlfors regular and splits ${\mathbb{R}}^{d+1}$ into two NTA domains then ${\omega }_{\Omega }\ll {\mathcal{H}}^d$ on $\Gamma \cap \partial \Omega$. This result is a natural generalisation of a result of Wu in [Wu86]. We also prove that almost every point in $\Gamma \cap \partial \Omega$ is a cone point if $\Gamma$ is a Lipschitz graph. Combining these results and a result from [AHMMMTV], we characterize sets of absolute continuity with finite ${\mathcal{H}}^d$-measure both in terms of the cone point condition and in terms of the rectifiable structure of the boundary. This generalizes the results of McMillan in [McM69] and Pommerenke in [Pom86]. Finally, we also show our first result holds for elliptic measure associated with real second order divergence form elliptic operators with a mild assumption on the gradient of the matrix.

Corrected several typos and errors