Global Calculus of Fourier Integral Operators, Weighted Estimates, and Applications to Global Analysis of Hyperbolic Equations

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The aim of this paper is to present certain global regularity properties of hyperbolic equations. In particular, it will be determined in what way the global decay of Cauchy data implies the global decay of solutions. For this purpose, global weighted estimates in Sobolev spaces for Fourier integral operators will be reviewed. We will also present elements of the global calculus under minimal decay assumptions on phases and amplitudes.

This work was completed with the aid of UK-Japan Joint Project Grant by The Royal Society and Japan Society for the Promotion of Science.