Let $n \in \mathbb{N}$, let $m \in \mathbb{R}$, and let $a: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{C}$ be a smooth symbol in the global Hörmander class $S^m_{1,0}(\mathbb{R}^n \times \mathbb{R}^n)$. Thus, for every pair of multi-indices $\alpha,\beta \in \mathbb{N}_0^n$, there exists a constant $C_{\alpha,\beta} > 0$ such that
for all $(x,\xi) \in \mathbb{R}^n \times \mathbb{R}^n$, where $\langle \xi\rangle := (1 + |\xi|^2)^{1/2}$.
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Let $\mathcal{F}: \mathcal{S}(\mathbb{R}^n) \to \mathcal{S}(\mathbb{R}^n)$ denote the [Fourier transform](/page/Fourier%20Transform) with symmetric normalization, and let $\mathcal{F}^{-1}$ denote its inverse. For $r \in \mathbb{R}$, let $H^r(\mathbb{R}^n)$ be the Bessel-potential [Sobolev space](/page/Sobolev%20Space) with norm
Let $A = \operatorname{Op}(a)$ be the left Kohn-Nirenberg pseudodifferential operator initially defined on $\mathcal{S}(\mathbb{R}^n)$ by the oscillatory integral