Fractional smoothness of distributions of polynomials and a fractional analog of the Hardy--Landau--Littlewood inequality

We prove that the distribution density of any non-constant polynomial $f(\xi_1,\xi_2,\ldots)$ of degree $d$ in independent standard Gaussian random variables $\xi$ (possibly, in infinitely many variables) always belongs to the Nikol'skii--Besov space $B^{1/d}(\mathbb{R}^1)$ of fractional order $1/d$ (and this order is best possible), and an analogous result holds for polynomial mappings with values in $\mathbb{R}^k$. Our second main result is an upper bound on the total variation distance between two probability measures on $\mathbb{R}^k$ via the Kantorovich distance between them and a suitable Nikol'skii--Besov norm of their difference. As an application we consider the total variation distance between the distributions of two random $k$-dimensional vectors composed of polynomials of degree $d$ in Gaussian random variables and show that this distance is estimated by a fractional power of the Kantorovich distance with an exponent depending only on $d$ and $k$, but not on the number of variables of the considered polynomials.