Function spaces between BMO and critical Sobolev spaces

Abstract

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 , p ( R n ) , where s p = 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.