Tangent measures and absolute continuity of harmonic measure
Jonas Azzam, Mihalis Mourgoglou · Revista Matemática Iberoamericana · 2018
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 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.