Let $\Omega \subset \mathbb{R}^n$ be a open and bounded domain with $\partial \Omega \in C^1$.
If $kp>n$ then $W^{k,p}(\Omega)$ is compactly embedded in $C^{k-k_0 -1 , \alpha}(\Omega)$ for any $\alpha<1-\beta$, where $k_0$ is the integer part of $n/p$ and $\beta$ the fractionary part.
This is one of the items from the Rellich-Kondrachov Embedding Theorem that was presented to me. The suggestion for the proof is to see theorem 2.5.1 and remark 2.5.2 of the book "Weakly Differentiable Functions" by Ziemer. But I couldn't adapt the ideas for this case. Does anyone have a suggestion on how to proof the embedding ?
By the Ascoli-Arzela theorem, you can prove that if $0<a<b\le 1$, then $C^{n, b}(\Omega)$ is compactly embedded in $C^{n, a}(\Omega)$. So it is enough to show that $W^{k, p}(\Omega)$ is continuously embedded in $C^{k-k_0-1,1-\beta}(\Omega)$. Since the domain is of class $C^1$, you can use Stein’s extension theorem to extend any function in $W^{k,p}(\Omega)$ to $W^{k,p}(R^n)$ with a bound on the norm. So now you are in the entire space and need to show that $W^{k,p}(R^n)$ is continuously embedded in $C^{k-k_0-1,1-\beta}(R^n)$. If $k=1$ this is Morrey’s embedding theorem. If $k>1$, the proof is long and boring. The idea is simple. Take a function $u$ in $W^{k,p}$ and let $v$ be one of its derivatives of order $k-1$. Then $v$ belongs to $W^{1,p}$. Depending on $p>n$, $p=n$, $p<n$ you can apply the three embedding theorems to $v$: Morrey’s embedding theorem, the critical case embedding theorem, or the Sobolev Gagliardo Nirenberg embedding theorem. In the case $p>n$, you can stop. In the other cases you conclude that $v$ belongs to $L^{p_*}$ when $p<n$ or any $L^q$ for $n\le q<\infty$ when $p=n$. Next you take a derivative of $u$ of order $k-2$. By what you proved for $v$, you know that the gradient of $w$ belongs to $L^q$ where $q>p$. So you have to apply again the three embedding theorems depending on $q<n$, $q=n$ or $q>n$. For the details see Theorem 12.55 in Sobolev book