Regularity of Fourier integral operators with amplitudes in general Hormander classes

We prove the global Lp-boundedness of Fourier integral operators that model the parametrices for hyperbolic partial differential equations, with amplitudes in classical Hormander classes S rho,delta m(Rn) for parameters 0 <= rho <= 1, 0 <= delta <1. We also consider the regularity of operators with amplitudes in the exotic class S0,<delta>m(Rn), 0 <= delta <1 and the forbidden class S<rho>,1m(Rn), 0 <= rho <= 1. Furthermore we show that despite the failure of the L2-boundedness of operators with amplitudes in the forbidden class S1,10(Rn), the operators in question are bounded on Sobolev spaces Hs(Rn) with s>0. This result extends those of Y. Meyer and E. M. Stein to the setting of Fourier integral operators