# Reference request: The compactness and compact embedding in Besov Space?

Let $\Omega\subset \mathbb R^N$ be open bounded with smooth boundary. Let $0<s<1$, $1\leq p<\infty$, and $1\leq \theta\leq\infty$. We denote by $B^{s,p,\theta}(\Omega)$ the Besov space. For definition of Besov space we refer to Leoni's book, Chapter 14, section 14.1. (Also this book by Adam, page 230, section 7.32.)

Theorem 14.29 in Leoni's book states the continuous imbedding theorem for Besov space. (For simplification, let's assume $p=1$.) We have $B^{s,1,\theta}(\Omega)$ continuous imbedded in $L^{\frac{N}{N-s}}(\Omega)$ for $1\leq \theta\leq \frac{N}{N-s}$.

We now take $r<\frac{N}{N-s}$.

My question is: do we have that $B^{s;1,\theta}(\Omega)$ is COMPACT imbedded in $L^{r}$? I think the answer is yes because according to this post, exercise 15, that

sequences bounded in a high regularity space, and constrained to lie in a compact domain, will tend to have convergent subsequences in low regularity spaces.

I would think my conjecture is true based on this fact. However, I did a deep search over the internet but has no lucky to find such result.

If there is no such result, please let me know (and maybe a counterexample?). If there is, please direct me to the reference.

Thank you!

• Have you looked into the book "Fourier Analysis and Nonlinear Partial Differential Equations" by Bahouri, Chemin, Danchin. In Section 2.9, Corollary 2.96 they prove that $B^{s,p,\infty}(K)$ is compactly embedded in $B^{s',p,1}(K)$ for $s'<s$ and compact $K$.
– gsa
Aug 5, 2015 at 7:31