Uniform domains with rectifiable boundaries and harmonic measure

Abstract

We assume that $\Omega \subset {\mathbb{R}}^{d+1}$, $d\ge 2$, 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.

arXiv comment15 pages, Minor typos and adjustments

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Cited by in corpus (3)

• Rectifiability of harmonic measure
• Rectifiability and elliptic measures on 1-sided NTA domains with Ahlfors-David regular boundaries
• Rectifiability, interior approximation and Harmonic Measure