On the distances between probability density functions

We give estimates of the distance between the densities of the laws of two functionals F and G on the Wiener space in terms of the Malliavin-Sobolev norm of F−G. We actually consider a more general framework which allows one to treat with similar (Malliavin type)methods functionals of a Poisson point measure (solutions of jump type stochastic equations). We use the above estimates in order to obtain a criterion which ensures that convergence in distribution implies convergence in total variation distance; in particular, if the functionals at hand are absolutely continuous, this implies convergence in L1 of the densities.