Harmonic Measure in Convex Domains

Introduction. Let Q be an open, convex subset of R^. At almost every point x of dQ, with respect to surface measure da, there is a unique outer unit normal 6. The map g: dQ -* S given by g(x) = 9 and defined almost everywhere is called the Gauss map. (S, n = N 1, is the unit sphere in R^.) Suppose that the origin 0 belongs to Q. Harmonic measure for fit at 0 is the probability measure co such that for all continuous functions ƒ on dQ, M(0) = f fdco Jan where u solves the Dirichlet problem: Au = 0 in Q and u = ƒ on dQ. Since Q is a Lipschitz domain, Dahlberg's theorem [4] implies that dco and da are mutually absolutely continuous. Thus we can define a measure fi on S by ju = g*a> or //(^) = w(^-(£')) for a l l ocs" 1 .