Let $X$ be a smooth manifold, let $u \in \mathcal{D}'(X)$, and for each $s \in \mathbb{R}$ let $WF^s(u) \subset T^*X \setminus 0$ denote the Sobolev wave front set of order $s$, defined by the elliptic cutoff characterization using properly supported order-zero pseudodifferential operators. Let $WF(u) \subset T^*X \setminus 0$ denote the smooth wave front set, defined by the corresponding elliptic cutoff characterization with smooth output. Fix an auxiliary Riemannian metric on $X$, let $S^*X$ denote the corresponding unit cosphere bundle, and let $\pi_S:T^*X\setminus 0\to S^*X$ be the radial projection. For a conic set $W\subset T^*X\setminus 0$, write $\pi_S(W)\subset S^*X$ for its image under radial projection. For $s\in\mathbb R$, write $H^s_{\mathrm{loc}}(X)$ for the local [Sobolev space](/page/Sobolev%20Space) of order $s$, and write $\Psi^0_{\mathrm{prop}}(X)$ for the class of properly supported pseudodifferential operators of order $0$ on $X$. Then

\begin{align*} WF(u)=\overline{\bigcup_{s\in\mathbb R} WF^s(u)}, \end{align*}

where the closure is taken in $T^*X \setminus 0$.

Equivalently, for every $(x_0,\xi_0) \in T^*X \setminus 0$, the following are equivalent:

1. $(x_0,\xi_0) \notin WF(u)$. 2. There exists an open conic neighbourhood $V \subset T^*X \setminus 0$ of $(x_0,\xi_0)$ such that

\begin{align*} V\cap WF^s(u)=\varnothing \end{align*}

for every $s\in\mathbb R$.