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

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- by Elena Cordero, Fabio Nicola and Luigi Rodino PDF
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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 $m=-d|1/2-1/p|$ 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.## References

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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