Showing that $H_{\text{loc}}^2(\mathbb{R}^2) \subset C^0(\mathbb{R}^2)$

Question

I want to show that $H_{\text{loc}}^2(\mathbb{R}^2) \subset C^0(\mathbb{R}^2)$

$C^0$ is the space of continuous functions, and $H_{\text{loc}}^2(\mathbb{R}^2)$ the set of distributions $u\in D'(\mathbb{R}^2)$ for which $\phi u \in H^2(\mathbb{R}^2)$ for all test-functions $\phi \in C_0^{\infty}(\mathbb{R}^2)$.

I appreciate all the help. Thanks.

Answer 1

Posting a CW to not leave the question unanswered.

1. Let $u\in H^2_{\rm loc}$. It suffices to prove that $u$ is continuous on $B_r=\{x:|x|<r\}$ for every $r$.
2. There is a test function $\phi$ such that $\phi=1$ on $B_r$
3. Since $\phi u\in H^2$, the Sobolev embedding implies that $\phi u$ is continuous. Therefore $u$ is continuous on $B_r=\{x:|x|<r\}$.