Let $\Omega$ be a open subset of $\mathbb{R}^{d}$ and $P \in \Psi^{m}(\Omega)$ a compactly supported pseudo-differential operator, that is, the kernel of $P$, is compact. Is it true that $P$ extends continuously from $H_{loc}^{s}(\Omega)$ into $H^{s-m}_{comp}(\Omega).$

I can show that $P:H_{loc}^{s}(\Omega) \longrightarrow H_{loc}^{s-m}(\Omega)$ and $P=\varphi P$, where $\varphi \in C_0^{\infty}(\Omega)$ and $\varphi=1$ in a neighborhood of $K'=\pi_x(\operatorname{supp}k_P)$, where $k_P$ denotes the (Schwartz) kernel of the operator $P$.

This is a strong indication that the statement is true.