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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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 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.

Article information

Source
Duke Math. J., Volume 163, Number 8 (2014), 1601-1654.

Dates
First available in Project Euclid: 26 May 2014

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1401146373

Digital Object Identifier
doi:10.1215/00127094-2713809

Mathematical Reviews number (MathSciNet)
MR3210969

Zentralblatt MATH identifier
1323.31008

Subjects
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]

Citation

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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