Higher integrability of the Harmonic Measure and Uniform Rectifiability

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Date(s) - 18 janvier 2012

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We present a higher dimensional, scale-invariant version of the classical theorem of F. and M. Riesz, which established absolute continuity of harmonic measure with respect to arc length measure, for a simply connected domain in the complex plane with a rectifiable boundary. More precisely, consider a domain, in dimension at least 3, with an Ahlfors-David regular boundary, which satifi-es the Harnack Chain condition plus an interior (but not exterior) corkscrew condition. We obtain that the harmonic measure is scale invariant absolutely continuous with respect to surface measure, along with higher integrability of the Poisson kernel, if and only if, the boundary of the domain is uniform rectifiable. Joint work with S. Hofmann, and with S. Hofmann and I. Uriarte-Tuero.