Vol. 10, No. 3, 2017

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

Steve Hofmann, Phi Le, José María Martell and Kaj Nyström

Vol. 10 (2017), No. 3, 513–558
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.

Keywords
harmonic measure and $p$-harmonic measure, Poisson kernel, uniform rectifiability, Carleson measures, Green function, weak-$A_\infty$
Mathematical Subject Classification 2010
Primary: 31B05, 31B25, 35J08, 42B25, 42B37
Secondary: 28A75, 28A78