We provide a simple abstract formalism of integration by parts under which we obtain
some regularization lemmas. These lemmas apply to any sequence of random variables
which are smooth and non-degenerated in some sense and enable one to upgrade the distance
of convergence from smooth Wasserstein distances to total variation in a quantitative way.
This is a well studied topic and one can consult for instance Bally and Caramellino [Electron. J. Probab. 2014],
Bogachev, Kosov and Zelenov [Trans. Amer. Math. Soc. 2018], Hu, Lu and Nualart [J. Funct. Anal. 2014],
Nourdin and Poly [Stoch. Proc. Appl. 2013] and the references therein for an overview of this issue.
Each of the aforementioned references share the fact that some non-degeneracy is required
along the whole sequence. We provide here the first result removing this costly assumption
as we require only non-degeneracy at the limit. The price to pay is to control the smooth
Wasserstein distance between the Malliavin matrix of the sequence and the Malliavin
matrix of the limit which is particularly easy in the context of Gaussian limit as their
Malliavin matrix is deterministic. We then recover, in a slightly weaker form, the main
findings of Nourdin, Peccati and Swan [J. Funct. Anal. 2014].
Another application concerns the approximation of the semi-group of a diffusion process
by the Euler scheme in a quantitative way and under the Hörmander condition.
Based on a common work with Lucia Caramellino and Guillaume Poly.