Vector-valued Sobolev spaces based on Banach function spaces. (October 2021)

• Record Type: Journal Article
• Authors: Evseev, Nikita
• Is Part Of: Nonlinear analysis. Volume 211(2021)
• Publication Date: 2021-10
• Subjects: 46E35 -- 46E40; Sobolev spaces -- Banach function space -- Vector-valued functions -- Reshetnyak–Sobolev space -- Newtonian space; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2021.112479
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: It is known that there are several approaches to define a Sobolev class for Banach valued functions. We compare the usual definition via weak derivatives with the Reshetnyak–Sobolev space and with the Newtonian space; in particular, we provide sufficient conditions when all three agree. Also, we revise the difference quotient criterion and the property of Lipschitz mapping to preserve Sobolev space when it is acting as a superposition operator.