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Funktsional'nyi Analiz i ego Prilozheniya, 2022, Volume 56, Issue 2, Pages 10–28
DOI: https://doi.org/10.4213/faa3988
(Mi faa3988)
 

This article is cited in 2 scientific papers (total in 2 papers)

Pointwise Conditions for Membership of Functions in Weighted Sobolev Classes

V. I. Bogachevabcd

a Lomonosov Moscow State University
b National Research University "Higher School of Economics", Moscow
c St. Tikhon's Orthodox University, Moscow
d Moscow Center for Fundamental and Applied Mathematics
Full-text PDF (647 kB) Citations (2)
References:
Abstract: According to a known characterization, a function $f$ belongs to the Sobolev space $W^{p,1}(\mathbb{R}^n)$ of functions contained in $L^p(\mathbb{R}^n)$ along with their generalized first-order derivatives precisely when there is a function $g\in L^p(\mathbb{R}^n)$ such that
$$ |f(x)-f(y)|\le |x-y|(g(x)+g(y)) $$
for almost all pairs $(x,y)$. An analogue of this estimate is also known for functions from the Gaussian Sobolev space $W^{p,1}(\gamma)$ in infinite dimension. In this paper the converse is proved; moreover, it is shown that the above inequality implies membership in appropriate Sobolev spaces for a large class of measures on finite-dimensional and infinite-dimensional spaces.
Keywords: Sobolev space, Gaussian measure, differentiable measure, quasi-invariant measure.
Received: 21.02.2022
Revised: 24.03.2022
Accepted: 25.03.2022
English version:
Functional Analysis and Its Applications, 2022, Volume 56, Issue 2, Pages 86–100
DOI: https://doi.org/10.1134/S0016266322020022
Bibliographic databases:
Document Type: Article
UDC: 517.518
Language: Russian
Citation: V. I. Bogachev, “Pointwise Conditions for Membership of Functions in Weighted Sobolev Classes”, Funktsional. Anal. i Prilozhen., 56:2 (2022), 10–28; Funct. Anal. Appl., 56:2 (2022), 86–100
Citation in format AMSBIB
\Bibitem{Bog22}
\by V.~I.~Bogachev
\paper Pointwise Conditions for Membership of Functions in Weighted Sobolev Classes
\jour Funktsional. Anal. i Prilozhen.
\yr 2022
\vol 56
\issue 2
\pages 10--28
\mathnet{http://mi.mathnet.ru/faa3988}
\crossref{https://doi.org/10.4213/faa3988}
\transl
\jour Funct. Anal. Appl.
\yr 2022
\vol 56
\issue 2
\pages 86--100
\crossref{https://doi.org/10.1134/S0016266322020022}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85139742301}
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  • https://doi.org/10.4213/faa3988
  • https://www.mathnet.ru/eng/faa/v56/i2/p10
  • This publication is cited in the following 2 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Функциональный анализ и его приложения Functional Analysis and Its Applications
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    Abstract page:327
    Full-text PDF :50
    References:53
    First page:26
     
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