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

## TL;DR

This paper proves that for certain sets with fractional Hausdorff measure, harmonic measure and Hausdorff measure are mutually singular, extending previous results by removing the uniform domain assumption.

## Contribution

It establishes the mutual singularity of harmonic measure and Hausdorff measure for sets with codimension less than one under a local capacity density condition, generalizing prior work.

## Key findings

- Harmonic measure cannot be mutually absolutely continuous with $H^s$ on the set $E$.
- The result holds without the uniform domain assumption.
- Answers a question posed by Azzam and Mourgoglou.

## 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.