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 $\hat{I}\text{c◯}={\mathbb{R}}^{n+1}\setminus E$ where $E$ is an $s$-Ahlfors regular compact set satisfying a uniform $L^2$-based non-flatness condition ${\hat{I}}_2^2\ge \hat{I}{´}_0$, the dimension of its harmonic measure is strictly less than $s$ for $s\in (n-c\hat{I}{´}_0^2,n]$. For planar domains, we establish an analogous quantitative threshold $s_0=1-c\hat{I}{´}_0^2$ under Azzam's uniform non-flatness condition ${\hat{I}}_{\infty }^2+{\hat{I}}_{\mathrm{hole}}^2\ge \hat{I}{´}_0$.

35 pages; v3: corrected a typo in Remark 1.4