Let $\Omega \subset \mathbb{R}^N$ be a smooth bounded domain. Is it true that the imersion $W^{2,p}(\Omega) \hookrightarrow C^{1}(\overline{\Omega})$ for $p > N$ is compact? I know that
$$
(1)\,\,\,\,\,\,\,\,\,\,\,W^{2,p}(\Omega) \hookrightarrow C^{1,1-\frac{N}{p}}(\overline{\Omega}), \quad \forall p > N.
$$
I suspect that the imersion $C^{1,1-\frac{N}{p}}(\overline{\Omega}) \hookrightarrow C^{1}(\overline{\Omega})$ is compact. However, I cound't find any reference with this fact. Does anyone know where I can find a reference with this?
The context: Let $L$ be a uniformly elliptic operators of second order and consider $M_0 > 0$ such that the problem $$ \begin{cases} (L + M)u = f, \quad \Omega\\ u = 0, \quad \partial \Omega \end{cases} $$ has a unique solution in $H^{1}_0(\Omega)$ for all $M > M_0$ and $f \in L^{2}(\Omega)$. Also, consider the linear and continous solution operator $T_M : L^2(\Omega) \rightarrow H^{1}_0(\Omega)$ associated to the problem above. Taking $f$ in the set $$ A := \{u \in C^{1}(\Omega) : u = 0, \partial \Omega\}, $$ we conclude that $f \in L^{p}(\Omega)$, for all $p \geq 1$. By theory of regularity, we also have $T_M(f) =: u_f \in W^{2,p}(\Omega)$, for all $p \geq 1$. So, $T_M(A) \subset W^{2,p}(\Omega)$. By (1), I also have $T_M(A) \subset A$. I need to show that $T_M : A \rightarrow A$ is compact. I appreciate any help.
Yes, the embedding is compact, provided the domain is smooth enough, see Gilbarg-Trudinger Sections 7.10 and 7.12. It is essentially a consequence of Ascoli-Arzelà theorem.