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.

15 pages, Minor typos and adjustments

References in corpus (2)

• Uniform Rectifiability and harmonic measure IV: Ahlfors regularity plus Poisson kernels in $L^p$ implies uniform rectifiability
• Rectifiability of harmonic measure in domains with porous boundaries