Let $\Omega$ be a compact manifold in $\mathbb R^2$. For $1 \leq p \lt 4/3$ can we claim that

$W^{1,p}(\Omega) \subset L^1(\Omega) \subset W^{-1,p}(\Omega)$

with the first inclusion being compact and the second one being continuous?

Note that $W^{-1,p}(\Omega)$ is identified with $W^{-1,(p')'}(\Omega)$ which is the dual space of $W^{1,p'}(\Omega)$

MOTIVATION: I want to use an Aubin-Lions-type lemma without the reflexivity assumption but I'm not sure about the mentioned inclusions.

Any help or hint is much appreciated.

Thanks in advance!


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