The variation and Kantorovich distances between distributions of polynomials and a fractional analog of the Hardy--Landau--Littlewood inequality

Vladimir I. Bogachev, Egor D. Kosov, Georgii I. Zelenov

Published 2016-02-16Version 1

Our first 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 certain Besov--Nikol'skii norm of their difference. Our second result asserts that the distribution of a non-constant polynomial of order $d$ (possibly, in infinitely many variables) with respect to a Gaussian measure always belongs to the Besov--Nikol'skii space $N^{1/d}$, and an analogous result holds for polynomial mappings with values in $\mathbb{R}^k$. Finally, we consider the total variation distance between the distributions of two random $k$-dimensional vectors composed of polynomials of order $d$ in Gaussian variables. 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.