Remarks on a Wiener type pseudodifferential algebra and Fourier integral operators
type: Item
instance of: scholarly article
title: Remarks on a Wiener type pseudodifferential algebra and Fourier integral operators
zbMATH Open document ID: 0905.35103
published in: Mathematical Research Letters
publication date: 1998-12-15T00:00:00Z
review text: In the paper [ibid. 1, 185-192 (1994)] \textit{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 \(L_{2}\).
Mathematics Subject Classification ID: 35S05
Mathematics Subject Classification ID: 47G30
Mathematics Subject Classification ID: 35S30
zbMATH DE Number: 1019786
zbMATH Keywords: symbols
zbMATH Keywords: Hörmander class
zbMATH Keywords: boundedness in \(L_2\)
author: Abdesslam Boulkhemair
reviewed by: Vladimir Rabinovich
MaRDI profile type: Publication
full work available at URL: mrl.1997.v4.n1.a6
OpenAlex ID: W1988710431
DOI: 10.4310/MRL.1997.V4.N1.A6
description: scientific article; zbMATH DE number 1019786