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

It is well-known that quantitative, scale invariant absolute continuity (more precisely, the weak-$A_{\mathrm{\infty }}$ property) of harmonic measure with respect to surface measure, on the boundary of an open set $ \Omega\subset $\mathbb{R}$^{n+1}$ with Ahlfors-David regular boundary, is equivalent to the solvability of the Dirichlet problem in $\mathrm{\Omega }$, with data in $L^p(\mathrm{\partial }\mathrm{\Omega })$ for some $p<\mathrm{\infty }$. In this paper, we give a geometric characterization of the weak-$A_{\mathrm{\infty }}$ property, of harmonic measure, and hence of solvability of the $L^p$ Dirichlet problem for some finite $p$. This characterization is obtained under background hypotheses (an interior corkscrew condition, along with Ahlfors-David regularity of the boundary) that are natural, and in a certain sense optimal: we provide counter-examples in the absence of either of them (or even one of the two, upper or lower, Ahlfors-David bounds); moreover, the examples show that the upper and lower Ahlfors-David bounds are each quantitatively sharp.