Sharp embeddings of Besov spaces involving only logarithmic smoothness

We use Kolyada's inequality and its converse form to prove sharp embeddings of Besov spaces B-p,r(0,beta) (involving the zero classical smoothness and a logarithmic smoothness with the exponent beta) into Lorentz-Zygmund spaces. We also determine growth envelopes of spaces B-p,r(0,beta). In distinction to the case when the classical smoothness is positive, we show that we cannot describe all embeddings in question in terms of growth envelopes. (c) 2008 Elsevier Inc. All rights reserved.