On the global boundedness of Fourier integral operators

Abstract

We consider a class of Fourier integral operators, globally defined on ${\mathbb{R}}^d$, with symbols and phases satisfying product type estimates (the so-called $SG$ or scattering classes). We prove a sharp continuity result for such operators when acting on the modulation spaces $M^p$. The minimal loss of derivatives is shown to be $d|1/2-1/p|$. This global perspective produces a loss of decay as well, given by the same order. Strictly related, striking examples of unboundedness on $L^p$ spaces are presented.

30 pages

Cited by in corpus (1)

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