Duke Mathematical Journal

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

Steve Hofmann, José María Martell, and Ignacio Uriarte-Tuero

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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 n2, for an Ahlfors–David regular domain ΩRn+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 Math. J., Volume 163, Number 8 (2014), 1601-1654.

First available in Project Euclid: 26 May 2014

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Primary: 31B05: Harmonic, subharmonic, superharmonic functions
Secondary: 35J08: Green's functions 35J25: Boundary value problems for second-order elliptic equations 42B99: None of the above, but in this section 42B25: Maximal functions, Littlewood-Paley theory 42B37: Harmonic analysis and PDE [See also 35-XX]


Hofmann, Steve; Martell, José María; Uriarte-Tuero, Ignacio. Uniform rectifiability and harmonic measure, II: Poisson kernels in $L^{p}$ imply uniform rectifiability. Duke Math. J. 163 (2014), no. 8, 1601--1654. doi:10.1215/00127094-2713809. https://projecteuclid.org/euclid.dmj/1401146373

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