Let $U \subseteq \mathbb{R}^n$ be open, let $m \in \mathbb{R}$, and let $P \in \Psi^m_{1,0}(U)$ be a properly supported pseudodifferential operator in the standard Hörmander class. Thus, in every coordinate patch, the local full symbol of $P$ belongs to $S^m_{1,0}$, and the Schwartz kernel $K_P \in \mathcal{D}'(U \times U)$ is properly supported over both factors of $U \times U$.

Then $P$ has a unique continuous extension

\begin{align*} P:\mathcal{D}'(U) \to \mathcal{D}'(U) \end{align*}

which agrees with the original operator on $C_c^\infty(U)$ and satisfies, for every $u \in \mathcal{D}'(U)$,

\begin{align*} \operatorname{sing\,supp}(Pu) \subseteq \operatorname{sing\,supp}u. \end{align*}