The Mutual Singularity of Harmonic Measure and Hausdorff Measure of Codimension Smaller than One
Xavier Tolsa · International Mathematics Research Notices · 2019
Abstract Let $\Omega \subset{\mathbb{R}}^{n+1}$ be open and let $E\subset \partial \Omega $ with $0<{\mathcal{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 the Hausdorff measure ${\mathcal{H}}^s$ 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.