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

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.