REGULARITY OF FOURIER INTEGRAL OPERATORS WITH AMPLITUDES IN GENERAL HÖRMANDER CLASSES

Abstract

We prove the global Lp-boundedness of Fourier integral operators that model the parametrices for hyperbolic partial differential equations, with amplitudes in classical Hörmander classes Sρ,δm(Rn) for parameters 0 ≤ ρ≤ 1 , 0 ≤ δ< 1. We also consider the regularity of operators with amplitudes in the exotic class S0,δm(Rn), 0 ≤ δ< 1 and the forbidden class Sρ,1m(Rn), 0 ≤ ρ≤ 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.