Tangent measures and absolute continuity of harmonic measure

Jonas Azzam, Mihalis Mourgoglou

    Research output: Contribution to journalArticleResearchpeer-review

    3 Citations (Scopus)

    Abstract

    © 2018 European Mathematical Society. We show that for uniform domains Ω ⊆ ℝd+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 ≤ 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.
    Original languageEnglish
    Pages (from-to)305-330
    JournalRevista Matematica Iberoamericana
    Volume34
    Issue number1
    DOIs
    Publication statusPublished - 1 Jan 2018

    Keywords

    • Absolute continuity
    • Capacity density condition
    • Harmonic measure
    • Non-tangentially accessible (NTA) domains
    • Tangent measures
    • Uniform domains
    • Wolff snowflakes

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