Malliavin calculus on the Wiener-Poisson space and its application to canonical SDE with jumps

We study the existence and smoothness of densities of laws of solutions of a canonical stochastic differential equation (SDE) driven by a Lévy process through the Malliavin calculus on the Wiener-Poisson space. Our assumption needed for the equation is very simple, since we are considering the canonical SDE. Assuming that the Lévy process is nondegenerate, we prove the existence of a smooth density in the case where the coefficients of the equation are nondegenerate. Our main result is stated in Theorem 1.1.