Boundedness of Fourier Integral Operators on ℱℒ^{𝓅} spaces

Cordero, Elena; Nicola, Fabio; Rodino, Luigi

doi:10.1090/S0002-9947-09-04848-X

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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