The Mutual Singularity of Harmonic Measure and Hausdorff Measure of Codimension Smaller than One

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Abstract

Let ω ℝn+1 be open and let E ω with 0 < Hs(E) < ∞, for some s ϵ (n, n + 1), satisfy a local capacity density condition. In this paper it is shown that the harmonic measure cannot be mutually absolutely continuous with the Hausdorff measure Hs on E. This answers a question of Azzam and Mourgoglou, who had proved the same result under the additional assumption that ω is a uniform domain.
Original languageEnglish
Pages (from-to)13783-13811
Number of pages29
JournalInternational Mathematics Research Notices
Volume2021
Issue number18
DOIs
Publication statusPublished - 1 Sept 2021

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