Harmonic measure and quantitative connectivity: geometric characterization of the $L^p$ solvability of the Dirichlet problem. Part II

Abstract

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. Together with other previous results by Hofmann and Martell, this implies that the weak-$A_{\infty }$ condition for harmonic measure holds if and only if $\partial \Omega$ is uniformly $n$-rectifiable and the weak local John condition is satisfied. This yields the first geometric characterization of the weak-$A_{\infty }$ condition for harmonic measure, which is important because of its connection with the Dirichlet problem for the Laplace equation.

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