The mutual singularity of harmonic measure and Hausdorff measure of codimension smaller than one

Abstract

Let $\Omega\subset\mathbb R^{n+1}$ be open and let $E\subset \partial\Omega$ with $0<H^s(E)<\infty$, for some $s\in(n,n+1)$, satisfy a local capacity density condition. In this paper it is shown that the harmonic measure cannot be mutually absolutely continuous with $H^s$ on $E$. This answers a question of Azzam and Mourgoglou, who had proved the same result under the additional assumption that $\Omega$ is a uniform domain.