Let $U \subset \mathbb{R}^n$ be open, let $m \in \mathbb{R}$, and write $\langle\xi\rangle=(1+|\xi|^2)^{1/2}$ for $\xi\in\mathbb{R}^n$. A function $A \in C^\infty(U \times U \times \mathbb{R}^n;\mathbb{C})$ is an **amplitude of order $m$** if for every pair of compact sets $K,L \subset U$ and all multi-indices $\beta,\gamma,\alpha \in \mathbb{N}_0^n$ there is a constant $C_{K,L,\alpha,\beta,\gamma} > 0$ with

\begin{align*} |\partial_x^\beta \partial_y^\gamma \partial_\xi^\alpha A(x,y,\xi)| \leq C_{K,L,\alpha,\beta,\gamma}\,\langle \xi\rangle^{m-|\alpha|}, \qquad (x,y,\xi) \in K \times L \times \mathbb{R}^n . \end{align*}

A function $a \in C^\infty(U\times\mathbb{R}^n;\mathbb{C})$ belongs to the symbol class $S^\ell(U\times\mathbb{R}^n)$ if for every compact $K\subset U$ and all $\beta,\alpha\in\mathbb{N}_0^n$ there is $C_{K,\alpha,\beta}>0$ with $|\partial_x^\beta\partial_\xi^\alpha a(x,\xi)|\leq C_{K,\alpha,\beta}\langle\xi\rangle^{\ell-|\alpha|}$ for all $(x,\xi)\in K\times\mathbb{R}^n$; set $S^{-\infty}(U\times\mathbb{R}^n)=\bigcap_{\ell\in\mathbb{R}}S^\ell(U\times\mathbb{R}^n)$, and define amplitudes of order $-\infty$ analogously.

For an amplitude $A$ of order $m$, the oscillatory operator $I_A:C_c^\infty(U) \to C^\infty(U)$ is

\begin{align*} I_A u(x)=(2\pi)^{-n}\int_U\int_{\mathbb{R}^n}e^{i(x-y)\cdot \xi}A(x,y,\xi)u(y)\,d\mathcal{L}^n(\xi)\,d\mathcal{L}^n(y), \end{align*}

interpreted in the standard oscillatory sense, and for $a\in S^m(U\times\mathbb{R}^n)$ the left quantization $\operatorname{Op}_L(a):C_c^\infty(U)\to C^\infty(U)$ is

\begin{align*} \operatorname{Op}_L(a)u(x)=(2\pi)^{-n}\int_U\int_{\mathbb{R}^n}e^{i(x-y)\cdot\xi}a(x,\xi)u(y)\,d\mathcal{L}^n(\xi)\,d\mathcal{L}^n(y). \end{align*}

Then there exists a left symbol $a \in S^m(U \times \mathbb{R}^n)$ such that the operator $I_A-\operatorname{Op}_L(a):C_c^\infty(U)\to C^\infty(U)$ is smoothing, i.e. has a Schwartz kernel in $C^\infty(U\times U)$. Moreover, writing $D_y^\alpha=(-i)^{|\alpha|}\partial_y^\alpha$ and

\begin{align*} b_j(x,\xi)=\sum_{|\alpha|=j}\frac{1}{\alpha!}\,\partial_\xi^\alpha D_y^\alpha A(x,y,\xi)\big|_{y=x}, \end{align*}

one has $b_j\in S^{m-j}(U\times\mathbb{R}^n)$ for every $j\in\mathbb{N}_0$, and for every $N\in\mathbb{N}$,

\begin{align*} a-\sum_{j=0}^{N-1}b_j\in S^{m-N}(U\times\mathbb{R}^n); \end{align*}

in particular $a(x,\xi)\sim \sum_{\alpha\in\mathbb{N}_0^n}\frac{1}{\alpha!}\partial_\xi^\alpha D_y^\alpha A(x,y,\xi)\big|_{y=x}$.