On hypoellipticity of generators of Lévy processes

Abstract

We give a sufficient condition on a Lévy measure μ which ensures that the generator L of the corresponding pure jump Lévy process is (locally) hypoelliptic, i.e., $\mathop {\mathrm {sing\,supp}}u\subseteq \mathop {\mathrm {sing\,supp}}Lu$ for all admissible u. In particular, we assume that $\mu|_{\mathbb {R}^{d}\setminus \{0\}}\in C^{\infty}(\mathbb {R}^{d}\setminus \{0\})$ . We also show that this condition is necessary provided that $\mathop {\mathrm {supp}}\mu$ is compact.