Let $n \in \mathbb{N}$, let $m \in \mathbb{R}$, and let $a \in S^m_{1,0}(\mathbb{R}^n \times \mathbb{R}^n)$, where $S^m_{1,0}(\mathbb{R}^n \times \mathbb{R}^n)$ is the class of smooth functions $a:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{C}$ whose derivatives satisfy the type $(1,0)$ estimates. For multi-indices $\alpha,\beta \in \mathbb{N}_0^n$, define the symbol seminorm

\begin{align*} p_{\alpha,\beta}^{(m)}(a) := \sup_{(x,\xi)\in \mathbb{R}^n \times \mathbb{R}^n} \langle \xi\rangle^{-m+|\beta|}\left|\partial_x^\alpha \partial_\xi^\beta a(x,\xi)\right|, \end{align*}

where $\langle \xi\rangle := (1+|\xi|^2)^{1/2}$.

Let $\mathcal{S}(\mathbb{R}^n)$ denote the [Schwartz space](/page/Schwartz%20Space) and let $\mathcal{S}'(\mathbb{R}^n)$ denote the space of [tempered distributions](/page/Tempered%20Distributions) on $\mathbb{R}^n$. Let $A=\operatorname{Op}(a)$ be the Kohn-Nirenberg pseudodifferential operator initially defined as a map $A:\mathcal{S}(\mathbb{R}^n)\to\mathcal{S}'(\mathbb{R}^n)$ by the oscillatory integral

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

for $u\in\mathcal{S}(\mathbb{R}^n)$. Then, for every $s\in \mathbb{R}$, $A$ extends uniquely to a bounded [linear map](/page/Linear%20Map)

\begin{align*} A:H^s(\mathbb{R}^n)\to H^{s-m}(\mathbb{R}^n). \end{align*}

Moreover, for every fixed $s\in \mathbb{R}$ there exist an integer $N=N(n,m,s)\in \mathbb{N}$ and a constant $C=C(n,m,s)>0$ such that

\begin{align*} \|A\|_{\mathcal{L}(H^s(\mathbb{R}^n),H^{s-m}(\mathbb{R}^n))}\leq C\sum_{|\alpha|+|\beta|\leq N}p_{\alpha,\beta}^{(m)}(a). \end{align*}