Quantitative Absolute Continuity of Harmonic Measure, and the Lp Dirichlet Problem
For a domain ? ? Rd, quantitative, scale-invariant absolute continuity (more precisely, the weak-A? property) of harmonic measure with respect to surface measure on ??, is equivalent to the solvability of the Dirichlet problem for Laplaces equation, with data in some Lp space on ??, with p < ?. Drawing an analogy to the famous Wiener criterion, which characterizes the domains in which the classical Dirichlet problem, with continuous boundary data, can be solved, it is of interest to find criteria for Lp solvability, thus allowing for singular boundary data. We shall review known results in this direction, in which (within the past 18 months) a rather complete picture has now emerged.