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

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.