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

Abstract:

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 $\xi _i$ (possibly, in infinitely many variables) always belongs to the Nikolskii–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 Nikolskii–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.