Revista Matemática Iberoamericana


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Volume 34, Issue 1, 2018, pp. 305–330
DOI: 10.4171/RMI/986

Published online: 2018-02-06

Tangent measures and absolute continuity of harmonic measure

Jonas Azzam[1] and Mihalis Mourgoglou[2]

(1) Universitat Autònoma de Barcelona, Bellaterra, Spain
(2) Universitat Autònoma de Barcelona, Bellaterra, Spain and Centre de Recerca Matemàtica, Barcelona, Spain

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.

Keywords: Harmonic measure, Wolff snowflakes, non-tangentially accessible (NTA) domains, uniform domains, capacity density condition, tangent measures, absolute continuity

Azzam Jonas, Mourgoglou Mihalis: Tangent measures and absolute continuity of harmonic measure. Rev. Mat. Iberoam. 34 (2018), 305-330. doi: 10.4171/RMI/986