Dimension Drop for Harmonic Measure on Ahlfors Regular Boundaries

Abstract

We provide quantitative estimates for the dimension drop of harmonic measure. We show that for a domain $\Omega = \mathbb{R}^{n+1} \setminus E$ where $E$ is an $s$-Ahlfors regular compact set satisfying a uniform $L^2$-based non-flatness condition $\beta_2 \ge \delta_0$, the dimension of its harmonic measure is strictly less than $s$ for $s \in (n - c\delta_0^2, n]$. For planar domains, we establish an analogous quantitative threshold $s_0 = 1 - c\delta_0^2$ under Azzam's uniform non-flatness condition $\beta_\infty + \beta_{\operatorname{hole}} \ge \delta_0$.