Regularity of Fourier integral operators with amplitudes in general Hörmander classes

Abstract

We prove the global L p-boundedness of Fourier integral operators that model the parametrices for hyperbolic partial differential equations, with amplitudes in classical Hörmander classes Sm ρ ,δ(Rn) for parameters 0 ≤ ρ ≤ 1, 0 ≤ δ < 1. We also consider the regularity of operators with amplitudes in the exotic class Sm 0,δ(Rn), 0 ≤ δ < 1 and the forbidden class Sm ρ ,1(Rn), 0 ≤ ρ ≤ 1. Furthermore we show that despite the failure of the L2-boundedness of operators with amplitudes in the forbidden class S0 1,1(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.