Dimension drop for harmonic measure on Ahlfors regular boundaries

Abstract

We show that given a domain $\Omega \subseteq {\mathbb{R}}^{d+1}$ with uniformly non-flat Ahlfors $s$-regular boundary and $s\ge d$, the dimension of its harmonic measure is strictly less than $s$.

Corrected typos and mistakes (in particular, a part of Lemma 4.1 was poorly written), made minor corrections to statements of Lemma 4.1 and Lemma 4.2, amended introduction slightly, added a figure. To appear in Potential Analysis