In Brezis's functional analysis(p.291), it says the following
$$W_0^{1,p}(\Omega) \subset L^2(\Omega) \subset W^{-1, p'}(\Omega) \quad \mbox{if} \quad 2N/(N+2) \le p \le 2$$
whether $\Omega$ is bounded or not bounded. As far as I know, most of Sobolev's inequalities or embeddings apply to bounded domains. Would you explain why the first inclusion holds? What Sobolev inequalities should we use to prove that?
The space $W^{1,p}_0$ is continuously embedded into $L^q$ for $\frac1q=\frac1p-\frac1N$ (Thm 9.9 in Brezis). The value of $q$ is larger than $2$ iff $\frac12\ge\frac1q=\frac1p-\frac1N$ or equivalently $p\ge \frac{2N}{N+2}$. Then we know: $W^{1,p}\subset L^q \cap L^p$, and since $p \le 2\le q$, $W^{1,p}\subset L^2$ by Hoelder inequality.