|
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
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
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
Linking options:
https://www.mathnet.ru/eng/faa3988https://doi.org/10.4213/faa3988 https://www.mathnet.ru/eng/faa/v56/i2/p10
|
Statistics & downloads: |
Abstract page: | 327 | Full-text PDF : | 50 | References: | 53 | First page: | 26 |
|