TY - JOUR AU - Chen, Jie Cheng AB - In this paper, we investigate the L2 boundedness of the Fourier integral operator Tø,a with smooth and rough symbols and phase functions which satisfy certain non-degeneracy conditions. In particular, if the symbol a∈L∞Sρm\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$a \in {L^\infty}S_\rho ^m$$\end{document}, the phase function ø satisfies some measure conditions and ∥∇ξkϕ(⋅,ξ)∥L∞≤C∣ξ∣ϵ−k\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Vert\nabla _\xi ^k\phi (\cdot,\xi){\Vert_{{L^\infty}}} \le C|\xi {|^{\epsilon - k}}$$\end{document} for all k ≥ 2,ξ ≠ 0, and some ϵ > 0, we obtain that Tø,a is bounded on L2 if m