About Sobolev embeddings and compactness

Question

For an open and bounded subset $\Omega\subset \mathbb{R}^n$ one can show that under certain conditions for the boundary there is an embedding $H_s(\Omega)=W_s^2(\Omega)\subset C(\Omega)$ for $s=\lfloor d/2\rfloor+1$ with $\Vert f\Vert_{\infty}\leq C \Vert f \Vert_{H_s}$. The reference I was reading now claims, that this embedding is compact. I do not see how this is part of the Sobolev embedding theorems, so I tried using Arzela-Ascoli without success. Is this result clear? Or is there any good reference?

Answer 1

Adams's book on Sobolev spaces should have this for general $W_s^p$ spaces. A "quick" proof, assuming that you already know that functions in $W^2_s(\Omega)$ can be extended to $W^2_s(\mathbb{R}^n)$, follows from Morrey's inequality, which embeds $W^p_s$ into a Hölder space $C^{0,\alpha}$, and from there it's just an application of Arzela-Ascoli.