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

We characterize local embeddings of Besov spaces Bp,r0,b involving only a slowly varying smoothness bb 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 Bp,r0,b and to show that we cannot describe all local embeddings of Besov spaces Bp,r0,b into Lorentz–Karamata spaces in terms of growth envelopes.