On L 2 $L^{2}$ -boundedness of Fourier integral operators

Abstract Let T a , φ $T_{a,\varphi }$ be a Fourier integral operator with symbol a and phase φ. In this paper, under the conditions a ( x , ξ ) ∈ L ∞ S ρ n ( ρ − 1 ) / 2 ( ω ) $a(x,\xi )\in L^{\infty }S^{n(\rho -1)/2}_{\rho }(\omega )$ and φ ∈ L ∞ Φ 2 $\varphi \in L^{\infty }\varPhi ^{2}$ , the authors show that T a , φ $T_{a,\varphi }$ is bounded from L 2 ( R n ) $L^{2}(\mathbb{R}^{n})$ to L 2 ( R n ) $L^{2}(\mathbb{R}^{n})$ .