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

TL;DR

This paper demonstrates that polynomial distributions of Gaussian variables have fractional smoothness in a specific function space and establishes bounds on total variation distance using Kantorovich distance, with applications to polynomial vector distributions.

Contribution

The paper proves the fractional smoothness of polynomial distributions in Nikol'skii--Besov spaces and derives new bounds relating total variation and Kantorovich distances for polynomial distributions.

Findings

Polynomial densities belong to Nikol'skii--Besov space $B^{1/d}$.

Total variation distance is bounded by a fractional power of Kantorovich distance.

Results are applicable to polynomial mappings in Gaussian spaces.

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 $\xi$ (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.