Let $N \in \mathbb N$, let $U \subset \mathbb R^N$ be open, let $\phi \in C^\infty(U;\mathbb R)$, and let $a \in C_c^\infty(U;\mathbb C)$. Define the critical set

\begin{align*} C := \{x \in U : \nabla \phi(x) = 0\} \end{align*}

and the compact set

\begin{align*} K := \operatorname{supp} a \cap C. \end{align*}

Assume that there is an open neighbourhood $V \subset U$ of $K$ such that $C \cap V$ is clean near $K$ in the following sense: every connected component $C_\ell$ of $C \cap V$ with $C_\ell \cap K \neq \varnothing$ is an embedded submanifold of $U$, only finitely many such components meet $K$, and for every $x \in C_\ell$,

\begin{align*} \ker D^2\phi_x = T_x C_\ell. \end{align*}

For each such component $C_\ell$, set

\begin{align*} e_\ell := \dim C_\ell, \qquad k_\ell := N - e_\ell. \end{align*}

Let $\phi_\ell \in \mathbb R$ be the constant value of $\phi$ on $C_\ell$. For $x \in C_\ell$, let $H_\ell(x)$ be the symmetric nondegenerate [bilinear form](/page/Bilinear%20Form) induced by $D^2\phi_x$ on the Euclidean normal quotient $\mathbb R^N / T_xC_\ell$. Let $\sigma_\ell$ be the signature of $H_\ell(x)$, which is independent of $x \in C_\ell$, and let $|\det H_\ell(x)|$ denote the absolute determinant computed in any [orthonormal basis](/page/Orthonormal%20Basis) of $\mathbb R^N / T_xC_\ell$.

Then, as $\lambda \to +\infty$,

\begin{align*} \int_U e^{i\lambda \phi(x)}a(x)\,d\mathcal L^N(x) \sim \sum_\ell e^{i\lambda\phi_\ell}\lambda^{-k_\ell/2}\sum_{j=0}^{\infty}\lambda^{-j}A_{j,\ell}. \end{align*}

Equivalently, for every $M \in \mathbb N$, subtracting the terms with $0 \le j < M$ leaves a remainder of the form

\begin{align*} \sum_\ell O(\lambda^{-k_\ell/2-M}). \end{align*}

The leading coefficient is

\begin{align*} A_{0,\ell} = (2\pi)^{k_\ell/2}e^{i\pi\sigma_\ell/4} \int_{C_\ell} a(x)|\det H_\ell(x)|^{-1/2}\,d\mathcal H^{e_\ell}(x). \end{align*}

The measure $\mathcal H^{e_\ell}$ is the Euclidean [Hausdorff measure](/page/Hausdorff%20Measure) induced on $C_\ell$, and the last integral is localized to $\operatorname{supp} a \cap C_\ell$.