Quantitative Absolute Continuity of Harmonic Measure and the Dirichlet Problem: A Survey of Recent Progress

Abstract

It is a well-known folklore result that quantitative, scale invariant absolute continuity (more precisely, the weak-A∞ property) of harmonic measure with respect to surface measure, on the bound¬ary 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 < ∞. Drawing an analogy to the famous Wiener criterion, which characterizes the domains in which the classical Dirichlet problem, with contin¬uous boundary data, can be solved, one may seek to characterize the open sets for which Lp solvability holds, thus allowing for singular boundary data. It has been known for some time that absolute continuity of harmonic measure is closely tied to rectifiability properties of ∂ Ω, but also that rectifiability alone is not sufficient to guarantee absolute continuity. In this note, we survey recent progress in this area, culminating in a geometric charac¬terization of the weak-A∞ property, and hence of solvability of the Lp Dirichlet problem for some finite p. This characterization, obtained under rather optimal background hypotheses, follows from a combination of the present author’s joint work with Martell, and the work of Azzam, Mourgoglou and Tolsa.