Nuclear Embeddings of Besov Spaces into Zygmund Spaces

Abstract

Let $d\in {\mathbb {N}}$ and let $\Omega$ be a bounded Lipschitz domain in ${\mathbb {R}}^d$. We prove that the embedding $I_d{:}B^d _{p,q}(\Omega ) \longrightarrow L_p (\log L)_a (\Omega )$ is nuclear if $a<-1$ and $1\le p,q\le \infty$, while if $-1<a<0$, $2<p<\infty$ and $p\le q \le \infty$ the embedding $I_d$ fails to be nuclear. Furthermore, if $a=-1$, the embedding $I_d{:}B^d _{\infty ,\infty }(\Omega ) \longrightarrow L_\infty (\log L)_{-1} (\Omega )$ is not nuclear.