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

Authors: Vladimir I. Bogachev, Egor D. Kosov and Georgii I. Zelenov
Journal: Trans. Amer. Math. Soc. 370 (2018), 4401-4432
MSC (2010): Primary 60E05, 60E15; Secondary 28C20, 60F99
Published electronically: February 1, 2018
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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.

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Additional Information

Vladimir I. Bogachev
Affiliation: Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119991 Russia – and – National Research University Higher School of Economics, Moscow, 101000 Russia

Egor D. Kosov
Affiliation: Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119991 Russia – and – National Research University Higher School of Economics, Moscow, 101000 Russia

Georgii I. Zelenov
Affiliation: Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119991 Russia – and – National Research University Higher School of Economics, Moscow, 101000 Russia

Keywords: Distribution of a polynomial, Nikolskii--Besov class, Hardy--Landau--Littlewood inequality, total variation norm, Kantorovich norm
Received by editor(s): March 21, 2016
Received by editor(s) in revised form: December 22, 2016
Published electronically: February 1, 2018
Additional Notes: This work has been supported by the Russian Science Foundation Grant 14-11-00196 at Lomonosov Moscow State University.
Article copyright: © Copyright 2018 American Mathematical Society

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