Let $X$ be a smooth manifold of dimension $n$ (Hausdorff and second countable), let $S \subset X$ be an embedded submanifold of dimension $k$ and codimension $\ell := n-k \ge 1$, and define the conormal bundle

\begin{align*} N^*S := \{(x,\xi) \in T^*X : x \in S,\ \xi|_{T_xS} = 0\}, \end{align*}

with $N^*S \setminus 0$ its complement of the zero section.

For a coordinate chart $U \subset X$, an integer $N \ge 1$, and an open subset $\Omega \subset U \times (\mathbb{R}^N \setminus 0)$ that is conic in $\theta$ (i.e. $(x,\theta) \in \Omega$ and $r>0$ imply $(x,r\theta) \in \Omega$), a function $\phi \in C^\infty(\Omega;\mathbb{R})$ is called a **nondegenerate conic phase** if it is positively homogeneous of degree $1$ in $\theta$, satisfies $d_{x,\theta}\phi \neq 0$ throughout $\Omega$, and the differentials $d_{x,\theta}(\partial_{\theta_1}\phi),\dots,d_{x,\theta}(\partial_{\theta_N}\phi)$ are linearly independent at every point of the critical set

\begin{align*} C_\phi := \{(x,\theta) \in \Omega : d_\theta\phi(x,\theta) = 0\}. \end{align*}

Write

\begin{align*} \lambda_\phi: C_\phi &\to T^*U \setminus 0, \end{align*}

\begin{align*} (x,\theta) &\mapsto (x,\, d_x\phi(x,\theta)). \end{align*}

The phase **parametrizes $N^*S$ over $U$** if $\lambda_\phi$ is an injective immersion onto an open conic subset of $N^*S \setminus 0$.

**Part 1 (Local existence).** For every $(x_0,\xi_0) \in N^*S \setminus 0$ there exist a coordinate neighbourhood $U \subset X$ of $x_0$, with coordinates $\varphi: U \to \mathbb{R}^k \times \mathbb{R}^\ell$, $\varphi(x) = (y,z)$, in which $S \cap U = \{z = 0\}$, such that the phase

\begin{align*} \phi_0: (\varphi(U)) \times (\mathbb{R}^\ell \setminus 0) &\to \mathbb{R}, \end{align*}

\begin{align*} (y,z,\zeta) &\mapsto z \cdot \zeta = \sum_{j=1}^\ell z_j \zeta_j, \end{align*}

is a nondegenerate conic phase that parametrizes $N^*S \cap (T^*U \setminus 0)$; in particular $(x_0,\xi_0) \in \lambda_{\phi_0}(C_{\phi_0})$.

**Part 2 (Oscillatory integrals are conormal).** Let $\phi$ be a nondegenerate conic phase on $\Omega \subset U \times (\mathbb{R}^N \setminus 0)$ parametrizing $N^*S$ over $U$, let $m \in \mathbb{R}$, and let $a \in S^m_{\mathrm{cl}}(U;\mathbb{R}^N)$ be a classical symbol of order $m$ with $\operatorname{supp} a \subset \Omega$ whose $x$-support $\{x : (x,\theta) \in \operatorname{supp} a \text{ for some } \theta\}$ has compact closure in $U$. Then the oscillatory integral

\begin{align*} u(\varphi) := \lim_{\varepsilon \downarrow 0} \int_U \int_{\mathbb{R}^N} e^{i\phi(x,\theta)}\, a(x,\theta)\, \chi(\varepsilon\theta)\, \varphi(x)\, d\mathcal{L}^N(\theta)\, d\mathcal{L}^n(x), \qquad \varphi \in C_c^\infty(U), \end{align*}

(with $\chi \in C_c^\infty(\mathbb{R}^N)$, $\chi \equiv 1$ near $0$) defines a distribution $u \in \mathcal{D}'(U)$ independent of $\chi$, the limit exists, $u$ is smooth on $U \setminus S$, $u$ is conormal to $S$, and

\begin{align*} \operatorname{WF}(u) \subset \lambda_\phi\big(C_\phi \cap \operatorname{supp} a\big) \subset N^*S \setminus 0. \end{align*}