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On hypoellipticity of generators of Lévy processes
Helmut Abels and Ryad Husseini
We give a sufficient condition on a Lévy measure $\mu$ which ensures that the generator $L$ of the corresponding pure jump Lévy process is (locally) hypoelliptic, i.e., the singular support of $u$ is contained in the singular support of $Lu$ for all admissible $u$. In particular, we assume that $\mu$ is smooth away from the origin. We also show that this condition is necessary provided that the support of $\mu$ is compact.