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Null sets of harmonic measure on NTA domains: Lipschitz approximation revisited

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Abstract

We show the David–Jerison construction of big pieces of Lipschitz graphs inside a corkscrew domain does not require surface measure be upper Ahlfors regular. Thus we can study absolute continuity of harmonic measure and surface measure on NTA domains of locally finite perimeter using Lipschitz approximations. A partial analogue of the F. and M. Riesz Theorem for simply connected planar domains is obtained for NTA domains in space. As one consequence every Wolff snowflake has infinite surface measure.

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  1. Department of Mathematics, University of Washington, Box 354350, Seattle, WA, 98195-4350, USA

    Matthew Badger

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  1. Matthew Badger
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Correspondence to Matthew Badger.

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The author was partially supported by NSF grants DMS-0838212 and DMS-0856687.

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Badger, M. Null sets of harmonic measure on NTA domains: Lipschitz approximation revisited. Math. Z. 270, 241–262 (2012). https://doi.org/10.1007/s00209-010-0795-1

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  • DOI: https://doi.org/10.1007/s00209-010-0795-1

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