Uniform rectifiability and harmonic measure I: Uniform rectifiability implies Poisson kernels in $L^p$

Steve Hofmann, José María Martell

Published 2012-02-17, updated 2015-07-08Version 3

We present a higher dimensional, scale-invariant version of a classical theorem of F. and M. Riesz. More precisely, we establish scale invariant absolute continuity of harmonic measure with respect to surface measure, along with higher integrability of the Poisson kernel, for a domain $\Omega\subset \mathbb{R}^{n+1},\, n\geq 2$, with a uniformly rectifiable boundary, which satisfies the Harnack Chain condition plus an interior (but not exterior) corkscrew condition. In a companion paper to this one [HMU], we also establish a converse, in which we deduce uniform rectifiability of the boundary, assuming scale invariant $L^q$ bounds, with $q>1$, on the Poisson kernel.