Let $n \in \mathbb{N}$, let $X$ be a smooth manifold of dimension $n$, let $k \in \{0,\dots,n\}$, let $Y \subset X$ be an embedded submanifold of codimension $k$, and let $m \in \mathbb{R}$. Define the conormal bundle of $Y$ in $X$ by

\begin{align*} N^*Y := \{(y,\xi) \in T^*X|_Y : \xi(v)=0 \text{ for every } v \in T_yY\}. \end{align*}

Let $N^*Y \setminus 0$ denote the complement of the zero section.

Use the following common order convention on both sides. In every adapted coordinate chart $(U,\varphi)$ with coordinates $x=(x',x'') \in \mathbb{R}^{n-k}\times\mathbb{R}^k$ such that $Y\cap U=\{x''=0\}$, a conormal distribution of order $m$ is locally, modulo a smooth [regular distribution](/page/Regular%20Distribution), a finite sum of oscillatory integrals of the form

\begin{align*} u(x)=(2\pi)^{-k}\int_{\mathbb{R}^k} e^{i x''\cdot\theta} a(x,\theta)\,d\mathcal{L}^k(\theta), \end{align*}

where the amplitude satisfies

\begin{align*} a \in S^{m-k/2+n/4}(U\times\mathbb{R}^k). \end{align*}

The Lagrangian distribution space $I^m(X,\Lambda)$ is normalized so that a nondegenerate phase with $N$ phase variables has amplitudes of order

\begin{align*} m-N/2+n/4. \end{align*}

If $k>0$, then the space of conormal distributions to $Y$ of order $m$ is exactly $I^m(X,N^*Y\setminus 0)$ as a subspace of $\mathcal{D}'(X)$, with smooth regular distributions included on both sides. If $k=0$, so that $Y=X$ and $N^*Y\setminus 0=\varnothing$, the same local definition has no nonzero phase variables and gives precisely the smooth regular distributions on $X$, equivalently the distributions locally represented by functions in $C^\infty(X)$.