On the global L 2-boundedness of Fourier integral operators with rough amplitude and phase functions

Abstract

Abstract In this paper, we study the L 2 {L^{2}} -boundedness of Fourier integral operators T ϕ , a {T_{\phi,a}} with rough amplitude a ⁢ ( x , η ) {a(x,\eta)} and rough phase functions ϕ ⁢ ( x , η ) {\phi(x,\eta)} , which satisfies the new class of rough non-degeneracy condition. In this study, under the conditions a ∈ L ∞ ⁢ S ρ m {a\in L^{\infty}S^{m}_{\rho}} , ϕ ∈ L ∞ ⁢ Φ 2 {{\phi}\in L^{\infty}\Phi^{2}} and some conditions of m, we show that T ϕ , a {T_{\phi,a}} is bounded from L 2 {L^{2}} to itself. Our main results extend and improve some known results about L 2 {L^{2}} , however, the method we use to prove these results is new.