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Article

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

Transactions of the American Mathematical Society. 2018. Vol. 370. No. 6. P. 4401-4432.

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}(R) 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.