Weak Harnack Inequality and H\"older Regularity for Symmetric Stable L\'evy Processes

Abstract

In this paper we consider weak Harnack inequality and H\"older regularity estimates for symmetric $\alpha$-stable L\'evy process in $\mathbb{R}^d$, $\alpha \in (0,2)$, $d\geq 2$. We consider a symmetric $\alpha$-stable L\'evy process $X$ for which a spherical part $\mu$ of the L\'evy 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 H\"older regularity results.