Bounded Trace Sobolev Embedding

Question

Just a sanity check here: all the references I've seen concerning Sobolev spaces discuss Sobolev trace embeddings of the form $$\|u\|_{L^q(\partial\Omega)}\le C\|u\|_{W^{1,p}(\Omega)}$$ where $\Omega$ is bounded and sufficiently smooth, and $q$ is given in term of $p$ and the dimension $n$, with finite $q<\infty$. Is the reason for only considering finite $q$ simply because the Sobolev spaces that are embedded in $L^\infty(\partial\Omega)$ coincide with the Sobolev spaces that are embedded in $L^\infty(\Omega)$?

Answer 1

If $p\le n$ then we have these trace theorem for some $q<\infty$. If $p>n$ then $W^{1,p}$ is embedded in spaces of (Hoelder) continuous functions, for which traces exist as restrictions to the boundary. Hence these cases are not discussed.