Sharp embeddings of Besov spaces with logarithmic smoothness

Abstract

We prove sharp embeddings of Besov spaces B with the classical smoothness a and a logarithmic smoothness a into Lorentz-Zygmund spaces. Our results extend those with a = 0, which have been proved by D. E. Edmunds and H. Triebel. On page 88 of their paper (Math. Nachr. 207 (1999), 79-92) they have written: "Nevertheless a direct proof, avoiding the machinery of function spaces, would be. desirable." In our paper we give such a proof even in a more general context. We cover both the sub-limiting and the limiting cases and we determine growth envelopes of Besov spaces with logarithmic smoothness