I have a basic question about the following claim made in a book I'm reading. The setting is the following.
Let $U$ by an open subset of $\mathbb{R}^n$ with compact closure, and $R\in\Psi^{-\infty}(U)$ be a pseudodifferential operator (operating on smooth $\mathbb{C}$-valued functions on $\mathbb{R}^n$) whose symbol is smoothing. Then $R$ extends to a continuous map between the Sobolev spaces $H_s$ to $H_t$ for any $s$ and $t$ in $\mathbb{R}$.
So far so good. Then it is claimed that $R(H_s)\subseteq C_c^\infty(U)$ by the Sobolev Lemma. I can see that the Sobolev Lemma gives $R(H_s)\subseteq C^\infty(U)$, but not why it should land in the compactly supported functions. Why is this the case?
I know I'm probably missing something simple here. Thanks for your help.
For any $f\in H_s$, $R(f)$ is compactly supported because of the definition of $R(f)$ as $\int_{U} e^{i x\cdot\xi}r(x,\xi)\hat{f}(\xi)d\xi$, where $r$ is the symbol of $R$. $R(f)$ inherits the support of $r(x,\xi)$. One sees that a limit $\lim_{n\rightarrow\infty} R(f_n)$ of such images must be supported similarly. The key here being that not only is each element in the image sequence compactly supported, they have the same support, hence their limit must too.