Question: can we claim that $W^{1,p}(\Omega) \subset L^1(\Omega) \subset W^{-1,p}(\Omega)$?

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.