Fractional smoothness of distributions of trigonometric polynomials on a space with a Gaussian measure

In this paper we study properties of images of a gaussian measure under trigonometric polynomials of a fixed degree, defined on finite-dimensional space with fixed number of dimensions. We prove that the images of the $n$-dimensional Gaussian measure under trigonometric polynomials have densities from the Nikolskii–Besov class of fractional parameter. This property of images of a gaussian measure is used for estimating the total variation distance between such images via the Fortet–Mourier distance. We also generalize these results to the case of $k$-dimensional mappings whose components are trigonometric polynomials.