Let $W \subset \mathbb{R}^n \times \mathbb{R}^n$ be open, let $m \in \mathbb{R}$, and let $A \in C^\infty(W \times \mathbb{R}^n)$ satisfy the following local symbol estimates of order $m$ and type $(1,0)$ in the $\xi$ variable: for every compact set $K \subset W$ and every pair of multi-indices $\alpha,\beta \in \mathbb{N}_0^n$ and every multi-index $\gamma \in \mathbb{N}_0^n$, there exists a constant $C_{K,\alpha,\beta,\gamma} > 0$ such that

\begin{align*} |D_x^\alpha D_y^\beta D_\xi^\gamma A(x,y,\xi)| \leq C_{K,\alpha,\beta,\gamma}(1+|\xi|)^{m-|\gamma|} \end{align*}

for all $(x,y) \in K$ and all $\xi \in \mathbb{R}^n$.

Let $r>0$ and let $W_r \subset W$ be an open subset such that $|x-y| \geq r$ for every $(x,y) \in W_r$. Fix a cutoff function $\rho \in C_c^\infty(\mathbb{R}^n)$ with $\rho(\xi)=1$ on a neighbourhood of $0$. Define the regularized maps $F_\varepsilon: W_r \to \mathbb{C}$, for $\varepsilon>0$, by

\begin{align*} F_\varepsilon(x,y) := \int_{\mathbb{R}^n} e^{i(x-y)\cdot \xi}\rho(\varepsilon \xi)A(x,y,\xi)\,d\mathcal{L}^n(\xi). \end{align*}

Define the candidate oscillatory kernel $F:W_r\to\mathbb{C}$ by

\begin{align*} F(x,y) := \lim_{\varepsilon \downarrow 0} F_\varepsilon(x,y) \end{align*}

whenever the limit exists.

Then the limit exists for every $(x,y) \in W_r$, is independent of the choice of such cutoff $\rho$, and defines a smooth function $F \in C^\infty(W_r)$. Moreover, for every multi-index pair $\alpha,\beta \in \mathbb{N}_0^n$,

\begin{align*} D_x^\alpha D_y^\beta F(x,y) = \operatorname{Os}\!\int_{\mathbb{R}^n} D_x^\alpha D_y^\beta\left(e^{i(x-y)\cdot \xi}A(x,y,\xi)\right)\,d\mathcal{L}^n(\xi) \end{align*}

on $W_r$, equivalently derivatives in $x$ and $y$ are obtained by differentiating the phase and the amplitude under the oscillatory integral.