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

Weighted Sobolev L2 estimates for a class of Fourier integral operators