Bilinear Fourier integral operators

We study the boundedness of bilinear Fourier integral operators on products of Lebesgue spaces. These operators are obtained from the class of bilinear pseudodifferential operators of Coifman and Meyer via the introduction of an oscillatory factor containing a real-valued phase of five variables \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Phi(x,y_1,y_2,\xi_1,\xi_2)}$$\end{document} which is jointly homogeneous in the phase variables (ξ1, ξ2). For symbols of order zero supported away from the axes and the antidiagonal, we show that boundedness holds in the local-L2 case. Stronger conclusions are obtained for more restricted classes of symbols and phases.