Let $m \in \mathbb{R}$, let $n \in \mathbb{N}$, and let $\mathbb{N}_0 := \{0,1,2,\dots\}$. Let $a: \mathbb{R}^n_x \times \mathbb{R}^n_\xi \to \mathbb{C}$ be a smooth function in the Hörmander symbol class $S^m_{1,0}(\mathbb{R}^n_x \times \mathbb{R}^n_\xi)$, meaning that for every pair of multi-indices $\gamma,\rho \in \mathbb{N}_0^n$,

\begin{align*} \sup_{(x,\xi)\in \mathbb{R}^n\times\mathbb{R}^n}\langle \xi\rangle^{-m+|\rho|}|\partial_x^\gamma\partial_\xi^\rho a(x,\xi)|<\infty, \end{align*}

where $\langle \xi\rangle := (1+|\xi|^2)^{1/2}$. Define the [Fourier transform](/page/Fourier%20Transform) of $u \in \mathcal{S}(\mathbb{R}^n)$ by

\begin{align*} \hat{u}: \mathbb{R}^n_\xi \to \mathbb{C}, \qquad \xi \mapsto (2\pi)^{-n/2}\int_{\mathbb{R}^n} e^{-i y\cdot \xi}u(y)\,d\mathcal{L}^n(y), \end{align*}

and define the Kohn-Nirenberg pseudodifferential operator associated to $a$ by

\begin{align*} \operatorname{Op}(a): \mathcal{S}(\mathbb{R}^n) \to C^\infty(\mathbb{R}^n), \qquad u \mapsto \operatorname{Op}(a)u, \end{align*}

with

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

for $x \in \mathbb{R}^n$. Then $\operatorname{Op}(a)u \in \mathcal{S}(\mathbb{R}^n)$ for every $u \in \mathcal{S}(\mathbb{R}^n)$, and the map

\begin{align*} \operatorname{Op}(a): \mathcal{S}(\mathbb{R}^n) \to \mathcal{S}(\mathbb{R}^n) \end{align*}

is continuous with respect to the [Schwartz space](/page/Schwartz%20Space) topology.