Dimension Drop for Harmonic Measure on Ahlfors Regular Boundaries

We show that given a domain Ω⊆ℝd+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\Omega }\subseteq \mathbb {R}^{d+1}$\end{document} with uniformly non-flat Ahlfors s-regular boundary with s ≥ d, the dimension of its harmonic measure is strictly less than s.