Let $X$ be a smooth manifold, let $m \in \mathbb{R}$, and let

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

be a properly supported pseudodifferential operator with $P \in \Psi^m(X)$. Let $u \in \mathcal{D}'(X)$. Here $\mathcal{D}'(X)$ denotes the space of distributions on $X$, $C^\infty(X)$ denotes the space of smooth functions on $X$, $\Psi^r(X)$ denotes the class of properly supported pseudodifferential operators of order $r$ on $X$, and $\Psi^{-\infty}(X)$ denotes the class of smoothing operators, equivalently properly supported operators with smooth Schwartz kernel mapping $\mathcal{D}'(X)$ into $C^\infty(X)$. With $WF(v) \subset T^*X \setminus 0$ denoting the wave front set of a distribution $v \in \mathcal{D}'(X)$ and $\operatorname{Char}(P) \subset T^*X \setminus 0$ denoting the characteristic set of $P$, one has

\begin{align*} WF(u) \setminus \operatorname{Char}(P) \subset WF(Pu). \end{align*}

Equivalently, for every $\rho_0 = (x_0,\xi_0) \in T^*X \setminus 0$, if $\rho_0 \notin \operatorname{Char}(P)$ and $\rho_0 \notin WF(Pu)$, then $\rho_0 \notin WF(u)$.