In Bresiz "Functional analysis" it is stated that if $I$ is a bounded interval in $\mathbb{R}$ then:
$W_0^{1,p}(I) \subset L^2(I) \subset W^{-1,p}(I)$.
(where $W_0^{1,p}$ is the closure of $C_0^{\infty}$ in $W^{1,p}$ and $W^{-1,p}$ is the dual space of $W^{1,p}$). It is also stated that the injections are contiuous if $1 \leq p < \infty$ and dense if $1 < p < \infty$.
My question is what this follows from and if (and under what conditions in that case) it holds for a domain $\Omega \subset \mathbb{R}^n$?