Smooth densities of stochastic differential equations forced by degenerate stable type noises

Using the Bismut's approach to Malliavin calculus, we introduce a simplified Malliavin matrix ([11]) for stochastic differential equations (SDEs) force by degenerate stable like noises. For the degenerate SDEs driven by Wiener noises, one can derive a Norris type lemma and use it \emph{iteratively} to prove the smoothness of density functions. Unfortunately, Norris type lemma is very hard to be iteratively applied to SDEs with stable like noises. In this paper, we derive a simple inequality as a replacement and use it to show that two families of degenerate SDEs with stable like noises admit smooth density functions. One family is the linear SDEs studied by Priola and Zabczyk ([13]), under some additional assumption we can iteratively use the inequality to get the smoothness of the density. The other family is the general SDEs with stable like noises, we can apply this inequality only \emph{one} time and thus derive that the SDEs admit smooth density if the \emph{first} order Lie brackets span $\R^d$. The crucial step in this paper is estimating the smallest eigenvalue of the simplified Malliavin matrix, which only uses some elementary facts of Poisson processes and undergraduate level ordinary differential equations.