Bilinear pseudodifferential operators on modulation spaces
type: Item
instance of: scholarly article
title: Bilinear pseudodifferential operators on modulation spaces
zbMATH Open document ID: 1061.47044
author: Árpád Bényi
author: Kasso A. Okoudjou
published in: The Journal of Fourier Analysis and Applications
publication date: 2005-02-11T00:00:00Z
review text: The boundedness of bilinear integral operators, in particular pseudo-differential operators with not necessarily smooth symbols, on products of modulation spaces is investigated. It is proved that if the kernel of the integral operator belongs to a modulation space \({\mathcal M}^1_{\Omega_s}({\mathbb R}^{3d})\), \(\Omega_s\) is a weight function of power type \(s\), then the operator is bounded from \({\mathcal M}^{p_1,p_2}_{\nu}({\mathbb R}^{d})\times {\mathcal M}^{q_1,q_2}_{\nu}({\mathbb R}^{d})\) into \({\mathcal M}^{r_1,r_2}_{\nu}({\mathbb R}^{d})\) if \(\frac{1}{r_1}=\frac{1}{p_1}+\frac{1}{q_1}\) and \(\nu\) is an \(s\)-moderated weight. A similar result holds for bilinear pseudodifferential operators \(T_\sigma\) with symbol \(\sigma\) belonging to a modulation space \({\mathcal M}^1_{\Omega^B_s}({\mathbb R}^{3d})\). In particular, if \(\sigma\) is an element of the Feichtinger algebra \({\mathcal M}^1({\mathbb R}^{3d})\), then \(T_\sigma\) is bounded from \(L^{p}({\mathbb R}^{d})\times L^{q}({\mathbb R}^{d})\) into \( L^{r}({\mathbb R}^{d})\) for \(2\leq p,q\leq \infty\), \(1\leq r\leq 2\) and \(\frac{1}{r}=\frac{1}{p}+\frac{1}{q}\). The key idea of the approach is the discretization of the modulation spaces by Gabor frames. In that way, the problem is reduced to the problem of infinite matrices acting on sequence spaces.
reviewed by: Leszek Skrzypczak
Mathematics Subject Classification ID: 47G30
Mathematics Subject Classification ID: 42B35
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 35S99
zbMATH DE Number: 2133426
zbMATH Keywords: bilinear operators
zbMATH Keywords: pseudodifferential operators
zbMATH Keywords: Gabor frames
zbMATH Keywords: modulation spaces
MaRDI profile type: Publication
full work available at URL: s00041-004-0977-5
OpenAlex ID: W2037309462
DOI: 10.1007/S00041-004-0977-5
description: scientific article; zbMATH DE number 2133426