A Seeger-Sogge-Stein theorem for bilinear Fourier integral operators

Abstract

We establish the regularity of bilinear Fourier integral operators with bilinear amplitudes in $S_{1,0}^m(n,2)$ and non-degenerate phase functions, from $L^p\times L^q\to L^r$ under the assumptions that $m\le -(n-1)(|\frac{1}{p}-\frac{1}{2}|+|\frac{1}{q}-\frac{1}{2}|)$ and $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$. This is a bilinear version of the classical theorem of Seeger-Sogge-Stein concerning the $L^p$ boundedness of linear Fourier integral operators. Moreover, our result goes beyond the aforementioned theorem in that it also includes the case of non-Banach target spaces.