On the global boundedness of Fourier integral operators

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 $d\mathrm{\mid }1\mathrm{/}2-1\mathrm{/}p\mathrm{\mid }$. 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.