## Duke Mathematical Journal

### Uniform rectifiability and harmonic measure, II: Poisson kernels in $L^{p}$ imply uniform rectifiability

#### 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 $n\geq2$, for an Ahlfors–David regular domain $\Omega\subset\mathbb {R}^{n+1}$ 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 $\partial \Omega$, with scale-invariant higher integrability of the Poisson kernel, is sufficient to imply quantitative rectifiability of $\partial \Omega$.

#### Article information

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

Dates
First available in Project Euclid: 26 May 2014

https://projecteuclid.org/euclid.dmj/1401146373

Digital Object Identifier
doi:10.1215/00127094-2713809

Mathematical Reviews number (MathSciNet)
MR3210969

Zentralblatt MATH identifier
1323.31008

#### 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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