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

• 成果类型: Article
• 署名作者: Azzam, Jonas; Hofmann, Steve; Maria Martell, Jose; Mourgoglou, Mihalis; Tolsa, Xavier
• 署名单位: University of Edinburgh; University of Missouri System; University of Missouri Columbia; Consejo Superior de Investigaciones Cientificas (CSIC); CSIC - Instituto de Ciencia de Materiales de Madrid (ICMM); CSIC - Instituto de Ciencias Matematicas (ICMAT); University of Basque Country; Basque Foundation for Science; ICREA; Autonomous University of Barcelona; Autonomous University of Barcelona
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-020-00984-5
• 发表日期: 2020
• 页码: 881-993
• 关键词: uniform rectifiability; carleson measures; Extrapolation; approximation

摘要

It is well-known that quantitative, scale invariant absolute continuity (more precisely, the weak-A(infinity) property) of harmonic measure with respect to surface measure, on the boundary of an open set Omega subset of Rn+1 with Ahlfors-David \regular boundary, is equivalent to the solvability of the Dirichlet problem in Omega, with data in L-p(partial derivative Omega) for some p < infinity. In this paper, we give a geometric characterization of the weak-A(infinity) property, of harmonic measure, and hence of solvability of the L-p 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 natural, and in a certain sense optimal: we provide counter-examples in the absence of either of them (or even one of the two, upper or lower, Ahlfors-David bounds); moreover, the examples show that the upper and lower Ahlfors-David bounds are each quantitatively sharp.