Weighted Sobolev L2 estimates for a class of Fourier integral operators

In this paper we develop elements of the global calculus of Fourier integral operators in $R^n$ under minimal decay assumptions on phases and amplitudes. We also establish global weighted Sobolev $L^2$ estimates for a class of Fourier integral operators that appears in the analysis of global smoothing problems for dispersive partial differential equations. As an application, we exhibit a new type of smoothing estimates for hyperbolic equations, where the decay of data in space is quantitatively translated into the time decay of solutions.