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 (involving the zero classical smoothness and a logarithmic smoothness with the exponent β) into Lorentz–Zygmund spaces. We also determine growth envelopes of spaces . 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.