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 $\alpha$-harmonic measure.\break The $\alpha$-harmonic measure is the hitting distribution of symmetric $\alpha$-stable processes upon exiting an open set in $\mathbb{R^n}$ ($0<\alpha<2$, $n\geq2$). It can also be defined in the context of Riesz potential theory and the fractional Laplacian. We prove polarization and symmetrization inequalities for $\alpha$-harmonic measure. We give a complete description of the corresponding equality cases. The proofs involve analytic and probabilistic arguments.