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 $\alpha$-stable Lévy process in ${\mathbb{R}}^d$, $\alpha \in (0,2)$, $d\ge 2$. We consider a symmetric $\alpha$-stable Lévy process $X$ for which a spherical part $\mu$ of the Lévy measure is a spectral measure. In addition, we assume that $\mu$ is absolutely continuous with respect to the uniform measure $\sigma$ 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.

The paper requires further work