Boundedness of Fourier integral operators on Fourier Lebesgue spaces and affine fibrations

TL;DR

This paper investigates the boundedness of Fourier integral operators on Fourier Lebesgue spaces, revealing that the loss of derivatives depends on the Hessian rank of the phase and the affine fibration structure.

Contribution

It establishes a precise relation between the Hessian rank of the phase function and the boundedness of Fourier integral operators on Fourier Lebesgue spaces, considering affine fibrations.

Findings

Operators of order -r|1/2-1/p| are bounded under certain affine fibration conditions.

The loss of derivatives is sharp and related to the Hessian rank of the phase.

Boundedness depends on the constancy of the gradient map on fibers of the affine fibration.

Abstract

We carry on the study of Fourier integral operators of H{\"o}rmander's type acting on the spaces $(\mathcal{F}L^p)_{comp}$, $1\leq p\leq\infty$, of compactly supported distributions whose Fourier transform is in $L^p$. We show that the sharp loss of derivatives for such an operator to be bounded on these spaces is related to the rank $r$ of the Hessian of the phase $\Phi(x,\eta)$ with respect to the space variables $x$. Indeed, we show that operators of order $m=-r|1/2-1/p|$ are bounded on $(\mathcal{F}L^p)_{comp}$, if the mapping $x\longmapsto\nabla_x\Phi(x,\eta)$ is constant on the fibers, of codimension $r$, of an affine fibration.