Function spaces between BMO and critical Sobolev spaces

Van Schaftingen, Jean [UCL]

The function spaces D-k(R-n) are introduced and studied. The definition of these spaces is based on a regularity property for the critical Sobolev spaces W-s,W-p(R-n), where sp = n, obtained by J.Bourgain, H.Brezis, New estimates for the Laplacian, the div-curl, and related Hodge systems, C. R. Math. Acad. Sci. Paris 338 (7) (2004) 539-543 (see also J. Van Schaftingen, Estimates for L-1-vector fields, C. R. Math. Acad. Sci. Paris 339 (3) (2004) 181-186). The spaces D-k(R-n) contain all the critical Sobolev spaces. They are embedded in BMO(R-n), but not in VMO(R-n). Moreover, they have some extension and trace properties that BMO(R-n) does not have. (c) 2006 Elsevier Inc. All rights reserved.