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

Abstract

Abstract. We carry on the study of Fourier integral operators of Hörmander's type acting on the spacesWe 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 onis constant on the fibers, of codimension r, of an affine fibration.

Estimates for rough Fourier integral and pseudodifferential operators and applications to the boundedness of multilinear operators

“…boundedness without the assumption of compact spatial support of the amplitude) has also been investigated in various contexts and here we would like to mention boundedness of operators with smooth amplitudes in the so called SG classes, due to E. Cordero, F. Nicola and L. Rodino in [8]; the boundedness of operators with amplitudes in S m 1,0 on the space of compactly supported distributions whose Fourier transform is in L p (R n ) (i.e. the F L p spaces) due to Cordero, Nicola and Rodino in [7] and Nicola's refinement of this investigation in [20]; and finally, S. Coriasco and M. Ruzhansky's global L p boundedness of Fourier integral operators [9], with amplitudes that belong to a certain subclass of S 0 1,0 . In this paper we consider the problem of boundedness of Fourier integral operators with amplitudes that are non-smooth in the spatial variables and exhibit an L p type behaviour in those variables for p ∈ [1, ∞].…”