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Remarks on a Wiener type pseudodifferential algebra and Fourier integral operators. (English) Zbl 0905.35103
In the paper [ibid. 1, 185-192 (1994)] J. Sjöstrand introduced a class of pseudodifferential operators with symbols defined without any references to derivatives. This class of symbols $$S_{w}$$ is defined in the following way: $$u:\mathbb {R}^{n}$$ $$\rightarrow \mathbb{C}$$ is in $$S_{w}$$ if for some $$\chi \in S(\mathbb {R}^{n})$$ with nonzero integral $\xi \rightarrow \sup_{\xi \in \mathbb {R}^{n}}\left| \mathcal{F}(u\tau _{k}\chi)(\xi)\right| \tag{1}$ is an integrable function in $$\mathbb{R}^{n}$$. Here, $$\mathcal{F}$$ denotes the Fourier transformation and $$\tau _{k}\chi (x)=\chi (x-k)$$. Equipped with the norm equal to the Lebesgue integral of (1), $$S_{w}$$ is a Banach space. The class $$S_{w}$$ contains the Hörmander class $$S_{0,0}^{0}$$. Moreover the operators in $$OPS_{w}$$ are bounded in $$L_{2}$$, and if $$A\in OPS_{w}$$ is invertible, then $$A^{-1}\in OPS_{w}$$. The author gives the following equivalent characteristic of the class $$S_{w}$$. He defines the class $$\mathcal{A}$$ of function $$u:\mathbb{R}^{n}$$ $$\rightarrow \mathbb{C}$$ such that for some $$\chi \in S(\mathbb{R}^{n})$$ with nonzero integral $k\rightarrow \sup_{x\in \mathbb{R}^{n}}\left| \mathcal{F}^{-1}[\mathcal{F} (u)\tau _{k}\chi)](x)\right| \tag{2}$ is an integrable function in $$\mathbb{R}^{n}$$. The author proves that the classes $$S_{w}$$ and $$\mathcal{A}$$ coincide. Moreover, he gives a very convenient proof for the boundedness in $$L_{2}$$ of the following symbols in $$S_{w}$$. Further, the author considers Fourier integral operators with amplitudes in $$S_{w}$$ and proves their boundedness in $$L_{2}$$.

##### MSC:
 35S05 Pseudodifferential operators as generalizations of partial differential operators 47G30 Pseudodifferential operators 35S30 Fourier integral operators applied to PDEs
##### Keywords:
symbols; Hörmander class; boundedness in $$L_2$$
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