Refined limiting imbeddings for Sobolev spaces of vector-valued functions

We prove a refined limiting imbedding theorem of the Brézis-Wainger type in the first critical case, i.e. s=Np, for Sobolev spaces Wps(RN,E) and Bessel potential spaces Hps(RN,E) of functions with values in a general Banach space E. In particular, the space E may lack the UMD property.