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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Boundedness of Fourier Integral Operators on $ \mathcal{F}L^p$ spaces


Authors: Elena Cordero, Fabio Nicola and Luigi Rodino
Journal: Trans. Amer. Math. Soc. 361 (2009), 6049-6071
MSC (2000): Primary 35S30, 47G30, 42C15
Published electronically: June 17, 2009
MathSciNet review: 2529924
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Abstract: We study the action of Fourier Integral Operators (FIOs) of Hörmander's type on $ \mathcal{F}L^p(\mathbb{R}^d)_{\operatorname{comp}}$, $ 1\le p\leq\infty$. We see, from the Beurling-Helson theorem, that generally FIOs of order zero fail to be bounded on these spaces when $ p\not=2$, the counterexample being given by any smooth non-linear change of variable. Here we show that FIOs of order $ m=-d\vert 1/2-1/p\vert$ are instead bounded. Moreover, this loss of derivatives is proved to be sharp in every dimension $ d\geq1$, even for phases which are linear in the dual variables. The proofs make use of tools from time-frequency analysis such as the theory of modulation spaces.


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Additional Information

Elena Cordero
Affiliation: Department of Mathematics, University of Torino, via Carlo Alberto 10, 10123 Torino, Italy
Email: elena.cordero@unito.it

Fabio Nicola
Affiliation: Dipartimento di Matematica, Politecnico di Torino, corso Duca degli Abruzzi 24, 10129 Torino, Italy
Email: fabio.nicola@polito.it

Luigi Rodino
Affiliation: Department of Mathematics, University of Torino, via Carlo Alberto 10, 10123 Torino, Italy
Email: luigi.rodino@unito.it

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04848-X
PII: S 0002-9947(09)04848-X
Keywords: Fourier Integral Operators, $\mathcal {F}L^p$ spaces, Beurling-Helson's theorem, modulation spaces, short-time Fourier transform
Received by editor(s): February 11, 2008
Published electronically: June 17, 2009
Article copyright: © Copyright 2009 American Mathematical Society