Tangent measures and absolute continuity of harmonic measure

We show that for uniform domains $Ω\subseteq \mathbb{R}^{d+1}$ whose boundaries satisfy a certain nondegeneracy condition that harmonic measure cannot be mutually absolutely continuous with respect to $α$-dimensional Hausdorff measure unless $α\leq d$. 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.

No reviews yet.

No comments yet.