Estimates and structure of α-harmonic functions

Abstract

We prove a uniform boundary Harnack inequality for nonnegative harmonic functions of the fractional Laplacian on arbitrary open set D. This yields a unique representation of such functions as integrals against measures on D c {∞} satisfying an integrability condition. The corresponding Martin boundary of D is a subset of the Euclidean boundary determined by an integral test. © 2007 Springer-Verlag.