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.