Uniform domains with rectifiable boundaries and harmonic measure

Mihalis Mourgoglou

Published 2015-05-22Version 1

We assume that $\Omega \subset \mathbb{R}^{d+1}$ is a uniform domain with lower $d$-Ahlfors-David regular rectifiable boundary so that $\mathcal{H}^d|_{\partial \Omega}$ is Radon. We show the Hausdorff measure $\mathcal{H}^d$ is absolutely continuous with respect to the harmonic measure $\omega$ in $\partial \Omega$ apart from a set of $\mathcal{H}^d$-measure zero.