The weak-$A_\infty$ property of harmonic and $p$-harmonic measures implies uniform rectifiability

TL;DR

This paper demonstrates that the weak-$A_ Infty$ property of harmonic and $p$-harmonic measures on Ahlfors-David regular sets implies their uniform rectifiability, linking measure-theoretic properties to geometric regularity.

Contribution

It establishes a new connection between the weak-$A_ Infty$ property of harmonic and $p$-harmonic measures and the uniform rectifiability of the underlying set.

Findings

Weak-$A_ Infty$ property implies uniform rectifiability for harmonic measure.

Generalizes the result to $p$-harmonic measure for $1<p< Infty$.

Links measure-theoretic properties with geometric regularity.

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 $E$. More generally, we establish a similar result for the Riesz measure, $p$-harmonic measure, associated to the $p$-Laplace operator, $1<p<\infty$.