Tangent measures and absolute continuity of harmonic measure

TL;DR

This paper investigates the relationship between harmonic measure and Hausdorff measure on boundaries of uniform domains, showing mutual absolute continuity only occurs when the Hausdorff dimension is at most the boundary's dimension.

Contribution

It introduces a tangent measure technique to analyze harmonic measure and establishes a dimension bound for absolute continuity on uniform domain boundaries.

Findings

Harmonic measure cannot be mutually absolutely continuous with Hausdorff measure unless the dimension is at most d.

A tangent measure lemma is developed for analyzing harmonic measure at nondegenerate points.

The results connect geometric boundary properties with measure-theoretic behavior of harmonic measure.

Abstract

We show that for uniform domains $\Omega\subseteq \mathbb{R}^{d+1}$ whose boundaries satisfy a certain nondegeneracy condition that harmonic measure cannot be mutually absolutely continuous with respect to $\alpha$-dimensional Hausdorff measure unless $\alpha\leq d$. We employ a lemma that shows that at almost every nondegenerate point, we may find a tangent measure of harmonic measure whose support is the boundary of yet another uniform domain whose harmonic measure resembles the tangent measure.