Dimension Drop for Harmonic Measure on Ahlfors Regular Boundaries

TL;DR

This paper establishes quantitative estimates showing that harmonic measure's dimension is strictly less than the boundary's Ahlfors regular dimension under certain non-flatness conditions, with specific thresholds depending on the domain's geometry.

Contribution

It provides new quantitative bounds on the dimension drop of harmonic measure for domains with Ahlfors regular boundaries satisfying uniform non-flatness conditions.

Findings

Harmonic measure dimension is less than boundary dimension under non-flatness conditions.

Explicit thresholds for dimension drop depending on non-flatness parameters.

Results apply to both higher-dimensional and planar domains with quantitative estimates.

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 $\beta_\infty + \beta_{\operatorname{hole}} \ge \delta_0$.