Let $W^{1,p}(\Omega)$ be the Sobolev space of weakly differentiable functions whose weak derivatives are $p$-integrable, where $\Omega \subset \mathbb R^n$ is a domain with Lipschitz boundary. Let furthermore $\gamma$ be the trace map.
I am looking for a theorem that allows me to conclude that $\gamma \colon W^{1,p}(\Omega) \to L^q(\partial \Omega)$ is compact whenever $q < \frac{(n-1)p}{n-p}$, thus e.g. for every $q < 4$ when $p = 2$ and $n = 3$.
(Necas, p103) has such a theorem in the necessary generality. I find the proof not very accessible, however. Another proof that I am aware of (Demengel/Demengel, p167) makes stronger assumptions on the regularity of the boundary ($C^1$ rather than $C^{0,1}$). Q1: Is such a theorem proved somewhere else in the same generality? Which source would you cite for said result?
Few books bother with the case of such embeddings as far as I can tell, most only consider embeddings of the type $W^{k,p}(\Omega) \to L^q(\Omega)$. Q2: Is that because the situation I am interested in is treated in the more general context of Besov spaces (which I am not familiar with) instead?
I have collected some more references for theorems of the Rellich-Kondrachov type here.
References:
Necas, Jindrich - Direct Methods in the Theory of Elliptic Equations.
Demengel, Françoise; Demengel, Gilbert - Functional spaces for the theory of elliptic partial differential equations.