Embeddings and the growth envelope of Besov spaces involving only slowly varying smoothness

We characterize local embeddings of Besov spaces B(p,r)(0,b) involving only a slowly varying smoothness b into classical Lorentz spaces. These results are applied to establish sharp local embeddings of the Besov spaces in question into Lorentz-Karamata spaces. As a consequence of these results, we are able to determine growth envelopes of spaces B(p,r)(0,b) and to show that we cannot describe all local embeddings of Besov spaces B(p,r)(0,b) into Lorentz-Karamata spaces in terms of growth envelopes. (C) 2011 Elsevier Inc. All rights reserved.