I am looking for a reference on the paper on compact Sobolev embeddings.  

If we define the Sobolev space $$X\_{0}(A):=\{u\in H^s(\mathbb R^N): u=0\quad \text{in}\quad \mathbb R^N \setminus A\}$$ where $A$ is an annulus and $s\in(0, 1)$. 


Is it true that the class of radial functions in $X\_{0}(A)$ is compact in $L^p(A)$ for any $p>1?$



compactness of fractional Sobolev spaces
You have the compact embedding of radial Sobolev functions in $X\_0(A)$ to $L^p(A)$ for all $1\leq p<\infty$ if and only if $s\geq 1/2$.


The proof goes as follows.


A radial function $F(x)$ on the annulus $A=\{ a<|x|<b\}$ is a function of the form $F(x)=f(|x|)$ for some $f$ defined on the interval $(a,b)$. Now $F\in L^p$ iff $f\in L^p(a,b)$ and $F\_k\to F$ in $L^p$ iff $f\_k\to f$ in $L^p(a,b)$ by Fubini's theorem. Also $F\in X\_0(A)$ iff $f\in H^s\_0(a,b)$,
Therefore compact embedding of radial functions in $X\_0(A)$ to all $L^p$, $1\leq p<\infty$ is equivalent to compact embedding of $H^s\_0(a,b)$ to $L^p(a,b)$ for all $1\leq p<\infty$. This however, is true when $s\geq 1/2$. Just check the textbooks that discuss compactness of embedding of $H^s$. I do not have a right reference on top of my head.


