Sobolev spaces of 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-Nikodým property

References in corpus (1)

• Mapping theorems for Sobolev-spaces of vector-valued functions