In the PDE book by Brezis, when $p = n$, Rellich-Kondrachov says that $W^{1, p}(\Omega) \subset \subset L^q(\Omega)$ is a compact injection for every $q \in [p, +\infty)$. Is the result still valid for $q \in [1, p)?$ If yes, where can I find a proof of this?
Rellich Kondrachov
partial differential equationsreference-request
Best Answer
Depends on whether $\Omega$ has finite measure.
If it does, then $L^q(\Omega) \subset L^{q'}(\Omega)$ whenever $q' \leq q$ by Hoelder. Then the result follows.
If $\Omega$ is sufficiently unbounded, then $W^{1,p}(\Omega)$ doesn't even embed into $L^q$ for $q < p$. So the result can't hold.