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Pointwise Conditions for Membership of Functions in Weighted Sobolev Classes

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Functional Analysis and Its Applications Aims and scope

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.

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Acknowledgments

The author is grateful to E. D. Kosov, S. N. Popova, and A. V. Shaposhnikov for useful discussions.

Funding

This research is supported by the Saint Tiknon’s Orthodox University and the Foundation “Living tradition,” by the Russian Foundation for Basic Research, Grant 20-01-00432, and by Moscow Center of Fundamental and Applied Mathematics.

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Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, Russia

    V. I. Bogachev

  2. National Research University Higher School of Economics, Moscow, Russia

    V. I. Bogachev

  3. Saint Tikhon’s Orthodox University, Moscow, Russia

    V. I. Bogachev

  4. Moscow Center for Fundamental and Applied Mathematics, Moscow, Russia

    V. I. Bogachev

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  1. V. I. Bogachev
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Correspondence to V. I. Bogachev.

Additional information

Translated from Funktsional'nyi Analiz i ego Prilozheniya, 2022, Vol. 56, pp. 10–28 https://doi.org/10.4213/faa3988.

Dedicated to the memory of Oleg Georgievich Smolyanov

Translated by V. I. Bogachev

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Bogachev, V.I. Pointwise Conditions for Membership of Functions in Weighted Sobolev Classes. Funct Anal Its Appl 56, 86–100 (2022). https://doi.org/10.1134/S0016266322020022

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