Weak Harnack Inequality and Hölder Regularity for Symmetric Stable Lévy Processes

Abstract

In this paper we consider weak Harnack inequality and Hölder regularity estimates for symmetric $α$-stable Lévy process in $\mathbb{R}^d$, $α\in (0,2)$, $d\geq 2$. We consider a symmetric $α$-stable Lévy process $X$ for which a spherical part $μ$ of the Lévy measure is a spectral measure. In addition, we assume that $μ$ is absolutely continuous with respect to the uniform measure $σ$ on the sphere and impose certain bounds on the corresponding density. Eventually, we show that the weak Harnack inequality holds, which we apply to prove Hölder regularity results.