  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.

Analysis of PDEs;Functional Analysis