Sobolev Embedding Theorems

Question

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.

Answer 1

The answer to the first question is yes. It can be found in many of the established texts. For the second question, some condition is needed at infinity. For instance, if the domain has a rapidly thinning "tentacle" extending to infinity, integral norms of derivatives can be finite without the function being finite. Hence a definition of "Lipschitz continuity" would have to exclude such domains.