Let $n \in \mathbb{N}$, and let $a: \mathbb{R}^n_x \times \mathbb{R}^n_\theta \to \mathbb{C}$ be a $C^\infty$ symbol in $S^0_{1,0}(\mathbb{R}^n_x \times \mathbb{R}^n_\theta)$, meaning that for every pair of multi-indices $\alpha,\beta \in \mathbb{N}_0^n$ there is a constant $C_{\alpha,\beta}>0$ such that $|\partial_x^\alpha\partial_\theta^\beta a(x,\theta)| \leq C_{\alpha,\beta}\langle \theta\rangle^{-|\beta|}$ for all $x,\theta \in \mathbb{R}^n$, where $\langle \theta\rangle = (1+|\theta|^2)^{1/2}$. Define the Kohn-Nirenberg pseudodifferential operator $\operatorname{Op}(a): \mathcal{S}(\mathbb{R}^n) \to \mathcal{S}'(\mathbb{R}^n)$ by the oscillatory integral $(\operatorname{Op}(a)u)(x) = (2\pi)^{-n}\int_{\mathbb{R}^n}\int_{\mathbb{R}^n} e^{i(x-y)\cdot \theta} a(x,\theta)u(y)\,d\mathcal{L}^n(y)\,d\mathcal{L}^n(\theta)$, interpreted by Schwartz cutoff regularization. Then there exist an integer $N=N(n)\in\mathbb{N}$ and a constant $C=C(n)>0$ such that $\|\operatorname{Op}(a)u\|_{L^2(\mathbb{R}^n)} \leq C \left(\max_{|\alpha|+|\beta|\leq N}\sup_{x,\theta \in \mathbb{R}^n} \langle \theta\rangle^{|\beta|}|\partial_x^\alpha\partial_\theta^\beta a(x,\theta)|\right)\|u\|_{L^2(\mathbb{R}^n)}$ for every $u \in \mathcal{S}(\mathbb{R}^n)$. Consequently $\operatorname{Op}(a)$ extends uniquely to a [bounded linear operator](/page/Bounded%20Linear%20Operator) on $L^2(\mathbb{R}^n)$.