On the global L¹-boundedness of Fourier integral operators with rough amplitude and phase functions

Let T_ϕ,a be a Fourier integral operator with amplitude a and phase functions ϕ. In this paper, we study the boundedness of Fourier integral operator of rough amplitude a ∈ L^∞S _ρ ^m and rough phase functions ϕ ∈ L^∞ Φ² with some measure condition. We prove the global L¹ boundedness for T_ϕ,a when \({1 \over 2} < \rho \leqslant 1\) $$\frac{1}{2}<\rho ⩽1$$ and \(m < \rho-{n+1\over 2}\) $$m<\rho -\frac{n+1}{2}$$ . Our theorem improves some known results.