Suppose $\Omega \subseteq \mathbb R^n$ ia a non-empty bounded domain with $C^{\infty}$-boundary and $p \in (1,\infty)$. Let $K \subseteq \Omega$ be a compact set. I'm aware of the usual Sobolev embedding theorems but I'm interested in what happens when restricting to compact sets. More precisely, (when) can we say that if $f\in W^{m,p}(\Omega)\cap W^{k,p}_0(\Omega)$, for some $m,k\in \mathbb N$, then $f\mid_{K}$ is bounded?
I would also be happy to look at any references which talks about similar results about compact sets.