Pointwise conditions for membership of functions in weighted Sobolev classes
type: Item
instance of: scholarly article
title: Pointwise conditions for membership of functions in weighted Sobolev classes
DOI: 10.1134/S0016266322020022
publication date: 2022-10-12T00:00:00Z
review text: It is known that the Sobolev space \(W^{1,p}(U)\) could be defined as a class of functions \(f\in L^p(U)\) so that for any element of this class there is a function \(g\in L^p(U)\) such that the following pointwise estimate \[ |f(x)-f(y)| \leq |x-y|(g(x)+g(y)) \] holds for almost all \(x,y\in U\). This characterization was employed to define Sobolev spaces on metric measure spaces [\textit{P. Hajłasz}, Potential Anal. 5, No. 4, 403--415 (1996; Zbl 0859.46022)]. Now, the author of the present paper considers the Sobolev space \(D^{1,p}_H(\mu)\) of functions defined on a Hilbert space \(H\), and with some measure \(\mu\). The question is whether the pointwise estimate defines such a Sobolev space. In particular, can we obtain the existence of the gradient from the estimate? The author gives an affirmative answer to such questions. In the case of the Gaussian Sobolev space \(W^{1,p}(\gamma)\), the main result reads as following. {Theorem 3.1.} Let \(\gamma\) be a centered Radon Gaussian measure on a locally convex space \(X\), and \(H\) be the Cameron-Martin space. Suppose that a \(\gamma\)-measurable function \(f\) satisfies the following condition: there is a sequence \((h_n)\), everywhere dense in \(H\), such that \[ |f(x+h_n) - f(x)| \leq |h_n|(g(x+h_n) - g(x))\text{ a.e.} \] where \(g\in L^p(\gamma)\). Then \(f\in W^{1,p}(\gamma)\) and \(|\nabla_H f(x)|_H \leq 2g(x)\) a.e.
reviewed by: Nikita Evseev
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 46G12
zbMATH DE Number: 7599891
zbMATH Keywords: Sobolev space
zbMATH Keywords: Gaussian measure
zbMATH Keywords: differentiable measure
zbMATH Keywords: quasi-invariant measure
zbMATH Keywords: Gaussian Sobolev space
zbMATH Keywords: Hajłasz-Sobolev space
author: Vladimir I. Bogachev
MaRDI profile type: Publication
cites work: Q4303140
cites work: Sobolev spaces on an arbitrary metric space
cites work: Monotone functions and quasiconformal mappings on Carnot groups
cites work: Grand Sobolev spaces on metric measure spaces
cites work: Nonsmooth calculus
cites work: Lusin-type approximation of Sobolev by Lipschitz functions, in Gaussian and \(\mathrm{RCD}(K,\infty)\) spaces
cites work: Q4219109
cites work: Q3579316
cites work: Q5250485
cites work: Ornstein–Uhlenbeck operators and semigroups
cites work: A note on Lusin-type approximation of Sobolev functions on Gaussian spaces
cites work: Q5642761
cites work: Analytic properties of infinite-dimensional distributions
cites work: Q3415147
cites work: \textit{BV} functions on convex domains in Wiener spaces
cites work: Sobolev functions on infinite-dimensional domains
cites work: \(BV\) functions on open domains: the Wiener case and a Fomin differentiable case
cites work: Q3265883
cites work: QUASI-INVARIANCE AND DIFFERENTIABILITY OF MEASURES
full work available at URL: s0016266322020022
OpenAlex ID: W4312806016
Recommended article: Sobolev spaces with respect to a weighted Gaussian measure in infinite dimensions
Recommended article: Characterisation of upper gradients on the weighted Euclidean space and applications
Recommended article: Q5687173
Recommended article: Q3142577
Recommended article: Finite-dimensional sections of functions from fractional Sobolev classes on infinite-dimensional spaces
Recommended article: A new characterization of Sobolev spaces
Recommended article: An integral characterization of Hajłasz-Sobolev space
Recommended article: Espaces de Sobolev gaussiens. (Gaussian Sobolev spaces)
Recommended article: Gradient estimates for perturbed Ornstein-Uhlenbeck semigroups on infinite-dimensional convex domains
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 7599891