Full paper in PDF:
$% P. Gurka and B. Opic, Sharp embeddings of Besov spaces with logarithmic smoothness, Rev. Mat. Complut. 18 (2005), no. 1, 81–110.%$

Sharp Embeddings of Besov Spaces with Logarithmic Smoothness

Petr GURKA and Bohumír OPIC
Department of Mathematics

Czech University of Agriculture
165 21 Prague 6 — Czech Republic

gurka@tf.czu.cz
Mathematical Institute

Academy of Sciences of the Czech Republic
Žitná 25
115 67 Prague 1 — Czech Republic

opic@math.cas.cz

Received: April 26, 2004
Accepted: June 17, 2004
ABSTRACT

We prove sharp embeddings of Besov spaces Bsp,a,r(Rn) with the classical smoothness s 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.

Key words: Besov spaces with logarithmic smoothness, Lorentz-Zygmund spaces, sharp embeddings.
2000 Mathematics Subject Classification: 46E35, 46E30, 26D10.