Symmetrization and harmonic measure

Abstract

We prove the equality statements for the classical symmetrization estimates for harmonic measure. In fact, we prove more general results for α-harmonic measure. The α-harmonic measure is the hitting distribution of symmetric α-stable processes upon exiting an open set in (0<α<2, n≥2). It can also be defined in the context of Riesz potential theory and the fractional Laplacian. We prove polarization and symmetrization inequalities for α-harmonic measure. We give a complete description of the corresponding equality cases. The proofs involve analytic and probabilistic arguments.