Vector-valued Sobolev spaces based on Banach function spaces

2020/09/21 by Evseev, Nikita
#46B22 #46E35 #46E40 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2009.09686

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.

Related

• Weak differentiability of metric space valued Sobolev maps
• Sobolev spaces of vector-valued functions on compact groups
• Vector calculus on weighted reflexive Banach spaces
• New Characterizations of First Order Sobolev Spaces
• Banach algebras of weakly differentiable functions