Regularity of density for SDEs driven by degenerate lévy noises

By using Bismut's approach to the Malliavin calculus with jumps, we study the regularity of the distributional density for SDEs driven by degenerate additive Lévy noises. Under full Hormander's conditions, we prove the existence of distributional density and the weak continuity in the first variable of the distributional density. Moreover, under a uniform first order Lie's bracket condition, we also prove the smoothness of the density.