Boundedness of Fourier integral operators on FL^p spaces

We study the action of Fourier Integral Operators (FIOs) of H¨ormander’s type on FL^p(R^d)_comp, 1 ≤ p≤∞. 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 ≥ 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.