Lipschitz functions and fractional Sobolev spaces
type: Item
instance of: scholarly article
title: Lipschitz functions and fractional Sobolev spaces
zbMATH Open document ID: 0944.46028
DOI: 10.1023/A:1008612206076
author: Francis Hirsch
published in: Potential Analysis
publication date: 2000-09-20T00:00:00Z
review text: The action of Lipschitz functions on fractional Sobolev spaces \(\mathbb{D}^\alpha_p\) corresponding to a sub-Markovian semigroup \((P_t)_{t\geq 0}\) is investigated. The semigroup defines the analytic strongly continuous semigroup of contractions in \(L^p(\mu)\), \(1<p<\infty\). The spaces \(\mathbb{D}^\alpha_p\), \(0<\alpha\leq 1\), are defined as Bessel potential spaces corresponding to the infinitesimal generator \(A_p\) of the semigroup in \(L^p(\mu)\). The examples are the heat semigroup with the classical Bessel potential spaces on \(\mathbb{R}^n\) and the Ornstein-Uhlenbeck semigroup on an abstract Wiener space with the Sobolev space involved in the Malliavin calculus. The author looks for conditions on the semigroup that guarantees that Lipschitz functions operate on the spaces by compositions. In particular, it is proved that for \(2\leq p<\infty\) the following assumptions imply the action of Lipschitz functions on \(\mathbb{D}^\alpha_p\): (0) the semigroup comes from a transition semigroup of a Markovian process, (1) measure \(\mu\) is a reference measure for the process, (2) there exists an algebra \({\mathcal D}\) of bounded functions contained in \(D(A_p)\) and dense in \(L^p\), (3) \(A_p\Gamma(f, f)- \Gamma(f, A_pf)- \Gamma(A_pf,f)\geq 0\) with \(\Gamma(f,g)= A_p(fg)- fA_pg- gAf\), \(f,g\in{\mathcal D}\). The assumptions are fulfilled by the both examples mentioned above.
reviewed by: Leszek Skrzypczak
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 47D07
Mathematics Subject Classification ID: 60H07
zbMATH DE Number: 1396420
zbMATH Keywords: Markovian semigroups
zbMATH Keywords: action of Lipschitz functions on fractional Sobolev spaces
zbMATH Keywords: sub-Markovian semigroup
zbMATH Keywords: analytic strongly continuous semigroup of contractions
zbMATH Keywords: infinitesimal generator
zbMATH Keywords: Bessel potential spaces
zbMATH Keywords: Ornstein-Uhlenbeck semigroup
zbMATH Keywords: abstract Wiener space
zbMATH Keywords: Malliavin calculus
zbMATH Keywords: Markovian process
MaRDI profile type: Publication
full work available at URL: a:1008612206076
OpenAlex ID: W1555045626
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description: scientific article; zbMATH DE number 1396420