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Uniform domains with rectifiable boundaries and harmonic measure (1505.06167v3)
Published 22 May 2015 in math.CA
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.