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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Boundedness of Fourier Integral Operators on $\mathcal {F}L^p$ spaces
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by Elena Cordero, Fabio Nicola and Luigi Rodino PDF
Trans. Amer. Math. Soc. 361 (2009), 6049-6071 Request permission

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 $m=-d|1/2-1/p|$ are instead bounded. Moreover, this loss of derivatives is proved to be sharp in every dimension $d\geq 1$, 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
  • MR Author ID: 629702
  • 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
  • MR Author ID: 149460
  • Email: luigi.rodino@unito.it
  • Received by editor(s): February 11, 2008
  • Published electronically: June 17, 2009
  • © Copyright 2009 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 361 (2009), 6049-6071
  • MSC (2000): Primary 35S30, 47G30, 42C15
  • DOI: https://doi.org/10.1090/S0002-9947-09-04848-X
  • MathSciNet review: 2529924