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

TL;DR

This paper proves that for domains with uniformly rectifiable boundaries satisfying certain geometric conditions, harmonic measure is absolutely continuous with respect to surface measure and the Poisson kernel has higher integrability, extending classical results.

Contribution

It establishes a scale-invariant absolute continuity of harmonic measure and higher integrability of the Poisson kernel for domains with uniformly rectifiable boundaries, under specific geometric conditions.

Findings

Harmonic measure is absolutely continuous with respect to surface measure.

Poisson kernel exhibits higher integrability in $L^p$ spaces.

Results extend classical theorems to higher dimensions and more general domains.

Abstract

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.