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

TL;DR

This paper investigates the boundedness of Fourier Integral Operators on Fourier Lebesgue spaces, establishing sharp conditions for their boundedness depending on the order and providing new insights using time-frequency analysis tools.

Contribution

It demonstrates that FIOs of order m = -d|1/2 - 1/p| are bounded on Fourier Lebesgue spaces and proves the sharpness of this result across all dimensions, even for linear phases.

Findings

FIOs of order zero are generally unbounded on these spaces for p ≠ 2.

FIOs of order m = -d|1/2 - 1/p| are bounded on Fourier Lebesgue spaces.

The sharpness of the boundedness condition is established in all dimensions.

Abstract

We study the action of Fourier Integral Operators (FIOs) of H{\"o}rmander's type on ${\mathcal{F}} L^p({\mathbb {R}}^d_{comp}$, $1\leq p\leq\infty$. 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 $d\geq1$, 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.