Tangent measures and absolute continuity 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 $\alpha \le d$. We employ a lemma that shows that, at almost every non-degenerate 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.