I am working on microlocal Sobolev spaces using Hörmander's books, and while trying to write down the details of a proof, I have faced the following problem. It looks easy, but I can't find a correct proof.
Let $\Omega$ be a subset of $\mathbb{R}^n$ and $u$ be a compactly supported distribution on $\Omega$. Let $P$ be a pseudo-differential operator on $\Omega$, of order $0$, and $\varphi \in \mathscr{C}^\infty_0(\Omega)$. Assume that there exists $s \in \mathbb{R}$ such that $$Pu \in H^s_{\mathrm{loc}}(\Omega).$$ I would like to show that $$P (\varphi u) \in H^s_{\mathrm{loc}}(\Omega).$$ If we change $H^s_{\mathrm{loc}}(\Omega)$ to $\mathscr{C}^\infty(\Omega)$, the problem looks simpler because of the pseudolocal property : we have $$\mathrm{sing} \ \mathrm{supp} \ Pu \subset \mathrm{sing} \ \mathrm{supp} \ u$$ but to conclude I would need $$\mathrm{sing} \ \mathrm{supp} \ P \phi u \subset \mathrm{sing} \ \mathrm{supp} \ Pu.$$ So even in this simple case I don't know how to conclude.