I am getting a little confused about the huge number of slight variations on the Sobolev Embedding Theorem.

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded Lipschitz domain and suppose that $f\in L_\infty(\Omega)\cap W^{\tau,2}(\Omega)$ for some $\tau\in\mathbb{R}$ with $\tau>n/2$. Do we have the inequality

$$ \left\Vert f\right\Vert_{L_\infty(\Omega)}\leq C\left\Vert f\right\Vert_{W^{\tau,2}(\Omega)} $$

for some constant C?

Does it hold if $\Omega$ is unbounded?

Thanks.