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 = N p , for Sobolev spaces Wsp (RN,E) and Bessel potential spaces Hsp (RN,E) of functions with values in a general Banach space E. In particular, the space E may lack the UMD property. © 2005 Elsevier Inc. All rights reserved.