Boundedness of Fourier Integral Operators on Fourier Lebesgue Spaces and Affine Fibrations

We carry on the study of Fourier integral operators of Hörmander’s type acting on the spaces FL(R)comp, 1 ≤ p ≤ ∞, of compactly supported distributions whose Fourier transform is in L. 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 Φ(x, η) with respect to the space variables x. Indeed, we show that operators of order m = −r|1/2 − 1/p| are bounded on FL(R)comp, if the mapping x 7−→ ∇xΦ(x, η) is constant on the fibers, of codimension r, of an affine fibration.