Uniform domains with rectifiable boundaries and harmonic measure

Abstract: We assume that $\Omega \subset \mathbb{R}^{d+1}$, $d \geq 2$, is a uniform domain with lower $d$-Ahlfors-David regular and $d$-rectifiable boundary. We show that if $\mathcal{H}^d|_{\partial \Omega}$ is locally finite, then the Hausdorff measure $\mathcal{H}^d$ is absolutely continuous with respect to the harmonic measure $\omega$ on $\partial \Omega$, apart from a set of $\mathcal{H}^d$-measure zero.