Bilinear Pseudodifferential Operators on Modulation Spaces

Bényi, Árpád; Okoudjou, Kasso A.

doi:10.1007/s00041-004-0977-5

Cited by 22 publications

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“…More precisely, we show that symbols in modulation spaces M p,q (R (m+1)d ), p ≥ q, give rise to bounded operators on products of corresponding modulation spaces; see the article of Czaja [12] and [17,18] for analogous results in the linear case. As a by-product of our analysis, we improve one of the main results in [4]. More precisely, we prove that if the symbol of a bilinear (or multilinear) pseudodifferential operator is in the Feichtinger algebra M 1 , then the corresponding operator maps…”

mentioning

confidence: 61%

“…It is sometimes denoted S 0 , and it has some remarkable properties; see [13] for a detailed description. In [4], we proved that M 1 as well as some of its weighted versions are convenient classes of symbols that give rise to bounded bilinear pseudodifferential operators on products of modulation spaces.…”

Section: 2mentioning

confidence: 99%

“…Indeed, it can be shown that symbols satisfying such conditions also belong to the modulation space M ∞,1 ; see [24,16,21]. We also wish to point out that the conditions on the symbols we employed in [4] and [2] are not comparable to the ones previously used in the "hard analysis" works cited above.…”

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confidence: 99%

“…More recently, in [4], we have initiated the study of multilinear pseudodifferential operators in the context of modulation spaces, which surprisingly appear as the right spaces in certain problems where more common L p estimates fail. These spaces play a dual role of classes of symbols and spaces of functions on which the operators act.…”

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confidence: 99%

“…These spaces play a dual role of classes of symbols and spaces of functions on which the operators act. For example, we proved in [4] that symbols in the so-called Feichtinger algebra yield bilinear operators that are bounded on products of modulation spaces. As a corollary, we also obtained boundedness results for operators in this class on products of Lebesgue and Sobolev spaces.…”

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confidence: 99%

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Modulation space estimates for multilinear pseudodifferential operators

Bényi

Okoudjou

2006

Studia Math.

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confidence: 61%

Section: 2mentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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confidence: 99%

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Modulation space estimates for multilinear pseudodifferential operators

Bényi

Okoudjou

2006

Studia Math.

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The Grossmann–Royer Transform, Gelfand–Shilov Spaces, and Continuity Properties of Localization Operators on Modulation Spaces

Teofanov

2018

Springer Proceedings in Mathematics &Amp; Statistics

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This paper offers a review of the results concerning localization operators on modulation spaces, and related topics. However, our approach, based on the Grossmann-Royer transform, gives a new insight and (slightly) different proofs. We define the Grossmann-Royer transform as interpretation of the Grossmann-Royer operator in the weak sense. Although such transform is essentially the same as the cross-Wigner distribution, the proofs of several known results are simplified when it is used instead of other time-frequency representations. Due to the importance of their role in applications when dealing with ultrafast decay properties in phase space, we give a detailed account on the Gelfand-Shilov spaces and their dual spaces, and extend the Grossmann-Royer transform and its properties in such context. Another family of spaces, modulation spaces, are recognized as appropriate background for time-frequency analysis. In particular, the Gelfand-Shilov spaces are revealed as projective and inductive limits of modulation spaces. For the continuity and compactness properties of localization operators we employ the norms in modulation spaces. We define localization operators in terms of the Grossmann-Royer transform, and show that such definition coincides with the usual definition based on the short-time Fourier transform.

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Continuity Properties of Multilinear Localization Operators on Modulation Spaces

Teofanov

2019

Applied and Numerical Harmonic Analysis

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We introduce multilinear localization operators in terms of the short-time Fourier transform, and multilinear Weyl pseudodifferential operators. We prove that such localization operators are in fact Weyl pseudodifferential operators whose symbols are given by the convolution between the symbol of the localization operator and the multilinear Wigner transform. For such interpretation we use the kenrel theorem for the Gelfand-Shilov space. Furthermore, we study the continuity properties of the multilinear localization operators on modulation spaces. Our results extend some known results when restricted to the linear case.1991 Mathematics Subject Classification. 47G30, 35S05, 46F05, 44A35.

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Bilinear Pseudodifferential Operators on Modulation Spaces

Cited by 22 publications

References 21 publications

Modulation space estimates for multilinear pseudodifferential operators

Modulation space estimates for multilinear pseudodifferential operators

The Grossmann–Royer Transform, Gelfand–Shilov Spaces, and Continuity Properties of Localization Operators on Modulation Spaces

Continuity Properties of Multilinear Localization Operators on Modulation Spaces

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