Characterizing Sobolev spaces of vector-valued functions

Abstract

We are concerned here with Sobolev-type spaces of vector-valued functions. For an open subset Ω⊂RN and a Banach space V, we characterize the functions in the Sobolev-Reshetnyak space R1,p(Ω, V), where 1 ≤p≤∞, in terms of the existence of partial metric derivatives or partial w∗-derivatives with suitable integrability properties. In the case p=∞ the Sobolev-Reshetnyak space R1,∞(Ω, V)is characterized in terms of a uniform local Lipschitz property. We also consider the special case of the space V=l∞.