On hypoellipticity of generators of Lévy processes
Helmut Abels and Ryad Husseini
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Submission date: 14. Aug. 2008
published in: Arkiv för matematik, 48 (2010) 2, p. 231-242
DOI number (of the published article): 10.1007/s11512-009-0099-z
MSC-Numbers: 35H10, 35S99, 47G20, 47G30, 60J75
Keywords and phrases: L\'evy processes, L\'evy measures, hypoelliptic, convolution operators, integro-differential operators
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We give a sufficient condition on a Lévy measure 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 is smooth away from the origin. We also show that this condition is necessary provided that the support of is compact.