Sobolev spaces of vector-valued functions

Abstract

We are concerned here with Sobolev-type spaces of vector-valued functions. For an open subset Ω ⊂ R N and a Banach space V , we compare the classical Sobolev space W1,p(Ω, V ) with the so-called Sobolev-Reshetnyak space R1,p(Ω, V ). We see that, in general, W1,p(Ω, V ) is a closed subspace of R1,p(Ω, V ). As a main result, we obtain that W1,p(Ω, V ) = R1,p(Ω, V ) if, and only if, the Banach space V has the Radon-Nikod´ym property.