Harmonic measure and quantitative connectivity: geometric characterization of the Lp-solvability of the Dirichlet problem

Jonas Azzam, Steve Hofmann, Jose Maria Martell, Mihalis Mourgoglou, Xavier Tolsa

Research output: Contribution to journalArticlepeer-review

Abstract

It is well-known that quantitative, scale invariant absolute continuity (more precisely, the weak-A∞ property) of harmonic measure with respect to surface measure, on the boundary of an open set Ω⊂R^n+1 with Ahlfors-David regular boundary, is equivalent to the solvability of the Dirichlet problem in Ω, with data in Lp(∂Ω) for some p<∞. In this paper, we give a geometric characterization of the weak-A∞ property, of harmonic measure, and hence of solvability of the Lp Dirichlet problem for some finite p. This characterization is obtained under background hypotheses (an interior corkscrew condition, along with Ahlfors-David regularity of the boundary) that are in the nature of best possible: we provide counter-examples in the absence of either of them (or even one of the two, upper or lower, Ahlfors-David bounds).
Original languageEnglish
Pages (from-to)881-993
JournalInventiones mathematicae
Volume222
Issue number3
Early online date20 Jul 2020
DOIs
Publication statusPublished - 31 Dec 2020

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