Uniform Rectifiability and harmonic measure IV: Ahlfors regularity plus Poisson kernels in $L^p$ implies uniform rectifiability

TL;DR

This paper proves that for Ahlfors-David regular sets, the weak-$A_ _ ext{infty}$ property of harmonic measure guarantees the set's uniform rectifiability, linking geometric and harmonic analysis properties.

Contribution

It establishes a new implication from harmonic measure properties to geometric regularity for Ahlfors-David regular sets.

Findings

Weak-$A_ extinfty$ property of harmonic measure implies uniform rectifiability

Connects harmonic measure behavior with geometric structure of sets

Extends understanding of rectifiability criteria in harmonic analysis

Abstract

Let $E\subset \mathbb{R}^{n+1}$, $n\ge 2$, be an Ahlfors-David regular set of dimension $n$. We show that the weak-$A_\infty$ property of harmonic measure, for the open set $\Omega:= \mathbb{R}^{n+1}\setminus E$, implies uniform rectifiability of $E$.