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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Fractional smoothness of distributions of polynomials and a fractional analog of the Hardy–Landau–Littlewood inequality
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by Vladimir I. Bogachev, Egor D. Kosov and Georgii I. Zelenov PDF
Trans. Amer. Math. Soc. 370 (2018), 4401-4432 Request permission

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
  • 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.
  • © Copyright 2018 American Mathematical Society
  • 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
  • MathSciNet review: 3811533