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

Steve Hofmann, José María Martell

Let $\Omega \subset {\mathbb{R}}^{n+1}$ be an open set, not necessarily connected, with an $n$-dimensional uniformly rectifiable boundary. We show that $\partial \Omega$ may be approximated in a…