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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
DOI: https://doi.org/10.1090/tran/7181
Published electronically: February 1, 2018
MathSciNet review: 3811533
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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
MR Author ID: 212251
Email: vibogach@mail.ru

Egor D. Kosov
Affiliation: Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119991 Russia
MR Author ID: 1020610
Email: ked_2006@mail.ru

Georgii I. Zelenov
Affiliation: Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119991 Russia
Email: zelenovyur@gmail.com

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