This post has been on MathOverflow for couple of days but receive no response. So I put it here hoping for more attentions.
Thank you guys!
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 $B^{s;1,\theta}(\Omega)$ 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.
So I would think my conjecture is true. 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 a reference.
Thank you!