Vector-valued Sobolev spaces based on Banach function spaces

TL;DR

This paper compares different definitions of Sobolev spaces for Banach-valued functions, establishing conditions for their equivalence and analyzing properties of Lipschitz mappings as superposition operators.

Contribution

It provides a comprehensive comparison of Sobolev space definitions and conditions for their equivalence, along with revising key criteria like difference quotients and Lipschitz mappings.

Findings

Conditions when weak, Reshetnyak-Sobolev, and Newtonian spaces coincide

Revised difference quotient criterion for Sobolev spaces

Analysis of Lipschitz mappings preserving Sobolev spaces

Abstract

It is known that for Banach valued functions there are several approaches to define a Sobolev class. 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. As well we revise the difference quotient criterion and the property of Lipschitz mapping to preserve Sobolev space when it acting as a superposition operator.