Uniform rectifiability and harmonic measure, II: Poisson kernels in Lp imply uniform rectifiability

Steve Hofmann, José María Martell, Ignacio Uriarte-Tuero

Type: Article

Publication Date: 2014-05-26

Citations: 40

DOI: https://doi.org/10.1215/00127094-2713809

Abstract

We present the converse to a higher-dimensional, scale-invariant version of the classical F. and M. Riesz theorem, proved by the first two authors. More precisely, for n≥2, for an Ahlfors–David regular domain Ω⊂Rn+1 which satisfies the Harnack chain condition plus an interior (but not exterior) corkscrew condition, we show that absolute continuity of the harmonic measure with respect to the surface measure on ∂Ω, with scale-invariant higher integrability of the Poisson kernel, is sufficient to imply quantitative rectifiability of ∂Ω.

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Duke Mathematical Journal - View
arXiv (Cornell University) - View - PDF
DIGITAL.CSIC (Spanish National Research Council (CSIC)) - View - PDF
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+ PDF Chat	A T(b) theorem with remarks on analytic capacity and the Cauchy integral	1990	Michael Christ
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+	On the Hausdorff dimension of harmonic measure in higher dimension	1987	Jean Bourgain
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