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

Abstract

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