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∞ property) of harmonic measure with respect to surface measure, on the boundary of an open set Ω ⊂ Rn+1 with Ahlfors–David regular boundary, is equivalent to the solvability of the Dirichlet problem in Ω , with data in Lp(∂Ω) for some p< ∞. In this paper, we give a geometric characterization of the weak-A∞ property, of harmonic measure, and hence of solvability of the Lp 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.