Sobolev spaces of vector-valued functions

TL;DR

This paper compares classical Sobolev spaces of vector-valued functions with Sobolev-Reshetnyak spaces, establishing conditions under which they coincide based on the Radon-Nikodým property of the Banach space.

Contribution

It characterizes when Sobolev and Sobolev-Reshetnyak spaces are equal for vector-valued functions, linking this to the Radon-Nikodým property of the target Banach space.

Findings

W^{1,p}(\Omega, V) is a closed subspace of R^{1,p}(\Omega, V)

Equality of the spaces occurs iff V has the Radon-Nikodým property

Provides a characterization of Sobolev spaces for vector-valued functions

Abstract

We are concerned here with Sobolev-type spaces of vector-valued functions. For an open subset $\Omega\subset\mathbb{R}^N$ and a Banach space $V$, we compare the classical Sobolev space $W^{1,p}(\Omega, V)$ with the so-called Sobolev-Reshetnyak space $R^{1,p}(\Omega, V)$. We see that, in general, $W^{1,p}(\Omega, V)$ is a closed subspace of $R^{1,p}(\Omega, V)$. As a main result, we obtain that $W^{1,p}(\Omega, V)=R^{1,p}(\Omega, V)$ if, and only if, the Banach space $V$ has the Radon-Nikod\'ym property