Let $X$ be a smooth manifold of dimension $n$, let $S \subset X$ be an embedded submanifold of dimension $k$ and codimension $\ell$, with $n=k+\ell$, and let $u \in \mathcal{D}'(X)$ be a conormal distribution to $S$ in the following local oscillatory-integral sense. First, $u$ is smooth on $X \setminus S$. Second, for every $p \in S$ there is an adapted coordinate chart $(U,\kappa)$, where $U \subset X$ is open, $V \subset \mathbb{R}^k \times \mathbb{R}^{\ell}$ is open, and $\kappa: U \to V$ is a diffeomorphism satisfying $\kappa(S \cap U)=V \cap (\mathbb{R}^k \times \{0\})$, such that for every cutoff $\chi \in C_c^\infty(U)$ with support contained in $U$, the pushed-forward localized distribution $(\kappa^{-1})^*(\chi u) \in \mathcal{D}'(V)$ is the sum of a smooth compactly supported function $w \in C_c^\infty(V)$ and an oscillatory integral distribution $v \in \mathcal{D}'(V)$. The distribution $v$ is defined by an amplitude $a: \mathbb{R}^k \times \mathbb{R}^{\ell} \to \mathbb{C}$ through the regularized pairing

\begin{align*} v(\phi)=\lim_{\varepsilon \downarrow 0}\int_{\mathbb{R}^k \times \mathbb{R}^{\ell} \times \mathbb{R}^{\ell}} e^{i z \cdot \theta}\rho(\varepsilon\theta)a(y,\theta)\phi(y,z)\,d\mathcal{L}^k(y)\,d\mathcal{L}^{\ell}(z)\,d\mathcal{L}^{\ell}(\theta) \end{align*}

for every $\phi \in C_c^\infty(V)$, where $\rho \in C_c^\infty(\mathbb{R}^{\ell})$ is any cutoff equal to $1$ near $0$, and the limit is independent of the choice of such $\rho$. The amplitude $a$ is smooth, compactly supported in $y$, and is a symbol of some fixed order $m \in \mathbb{R}$ in the variable $\theta$: for every pair of multi-indices $\alpha \in \mathbb{N}_0^k$ and $\beta \in \mathbb{N}_0^{\ell}$ there is a constant $C_{\alpha,\beta}>0$ such that

\begin{align*} |\partial_y^\alpha \partial_\theta^\beta a(y,\theta)| \le C_{\alpha,\beta}(1+|\theta|)^{m-|\beta|}. \end{align*}

Then

\begin{align*} \operatorname{WF}(u) \subset N^*S \setminus 0, \end{align*}

where

\begin{align*} N^*S := \{(p,\xi) \in T^*X : p \in S \text{ and } \xi(v)=0 \text{ for every } v \in T_pS\} \end{align*}

is the conormal bundle of $S$, and $0 \subset T^*X$ denotes the zero section.