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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
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