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

@article{Hofmann2020HarmonicMA, title={Harmonic measure and quantitative connectivity: geometric characterization of the \$\$L^p\$\$-solvability of the Dirichlet problem}, author={Steve Hofmann and Jos'e Mar'ia Martell}, journal={Inventiones mathematicae}, year={2020} }

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…

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