Let $n \in \mathbb{N}$, let $U \subset \mathbb{R}^n$ be open, let $K \subset U$ be compact, and let $\phi \in C^\infty(U;\mathbb{R})$. Assume that there exists $c > 0$ such that $|\nabla \phi(x)| \geq c$ for every $x \in K$. Let $a \in C_c^\infty(U;\mathbb{C})$ satisfy $\operatorname{supp} a \subset K$, and let $h_0 > 0$. Then for every $N \in \mathbb{N}$ there exists a constant $C_N > 0$ such that

\begin{align*} \left|\int_{\mathbb{R}^n} e^{i\phi(x)/h} a(x)\,d\mathcal{L}^n(x)\right| \leq C_N h^N \end{align*}

for every $h \in (0,h_0]$. The constant $C_N$ depends only on $N$, $h_0$, $c$, finitely many derivatives of $a$ on $K$, finitely many derivatives of $\phi$ on a compact neighbourhood of $K$ contained in $U$, and that neighbourhood.