Smoothness of Densities of Generalized Locally Non-Degenerate Wiener Functionals

Several criteria for existence of smooth densities of Wiener functionals are known in the framework of Malliavin calculus. In this article, we introduce the notion of generalized locally non-degenerate Wiener functionals and prove that they possess smooth densities. The result presented here unifies the earlier works by Shigekawa and Florit-Nualart. As an application, we prove that the law of the strong solution to a stochastic differential equation driven by Brownian motion admits a smooth density without an assumption of Lipschitz continuity for dispersion coefficients.