Harmonic measure and Riesz transform in uniform and general domains

Mihalis Mourgoglou, Xavier Tolsa

Research output: Contribution to journalArticleResearchpeer-review

Abstract

Let Ω ⊈ ℝn+1 be open and let μ be some measure supported on ∂Ω such that μ.B(x, r)) ≤ C rn for all x ∊ Rn+1, r > 0. We show that if the harmonic measure in Ω satisfies some scale invariant A1-type conditions with respect to μ, then the n-dimensional Riesz transform (equcation presented) is bounded in L2(μ). We do not assume any doubling condition on μ. We also consider the particular case when Ω is a bounded uniform domain. To this end, we need first to obtain sharp estimates that relate the harmonic measure and the Green function in this type of domains, which generalize classical results by Jerison and Kenig for the well-known class of NTA domains.
Original languageEnglish
Pages (from-to)183-221
Number of pages39
JournalJournal fur die Reine und Angewandte Mathematik
Volume2020
Issue number758
DOIs
Publication statusPublished - 1 Jan 2021

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