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

TL;DR

This paper proves that in certain high-dimensional domains, the absolute continuity of harmonic measure with a scale-invariant integrable Poisson kernel guarantees the boundary's uniform rectifiability, extending classical results.

Contribution

It establishes the converse relationship between harmonic measure properties and uniform rectifiability in higher dimensions, under specific geometric conditions.

Findings

Absolute continuity of harmonic measure implies uniform rectifiability.

Scale-invariant higher integrability of the Poisson kernel is sufficient.

Results extend classical theorems to higher-dimensional, scale-invariant settings.

Abstract

We present the converse to a higher dimensional, scale-invariant version of a classical theorem of F. and M. Riesz. More precisely, for $n\geq 2$, for an ADR domain $\Omega\subset \re^{n+1}$ which satisfies the Harnack Chain condition plus an interior (but not exterior) Corkscrew condition, we show that absolute continuity of harmonic measure with respect to surface measure on $\partial\Omega$, with scale invariant higher integrability of the Poisson kernel, is sufficient to imply uniformly rectifiable of $\partial\Omega$.