A geometric characterization of the weak-$A_\infty$ condition for harmonic measure

Jonas Azzam, Mihalis Mourgoglou, Xavier Tolsa

Published 2018-03-21Version 1

Let $\Omega\subset\mathbb R^{n+1}$ be an open set with $n$-AD-regular boundary. In this paper we prove that if the harmonic measure for $\Omega$ satisfies the so-called weak-$A_\infty$ condition, then $\Omega$ satisfies a suitable connectivity condition, namely the weak local John condition. This yields the first geometric characterization of the weak-$A_\infty$ condition of harmonic measure, which is important because its connection with the Dirichlet problem for the Laplace equation.