Bilinear Fourier integral operator and its boundedness

We consider the bilinear Fourier integral operatorS_\sigma(f,g)=\int_{\mathbb{R}^d}\int_{\mathbb{R}^d}e^{i\phi_۱(x,\xi)}e^{i\phi_۲(x,\eta)}\sigma(x,\xi,\eta)\hat{f}(\xi)\hat{g}(\eta)d\xi d\etaon modulation spaces. Our aim is to indicate this operator is well defined on S(\mathbb{R}^d) and shall show the relationship between the bilinear operator and BFIO on modulation spaces.

Keywords:

Fourier integral operator , boundedness , modulation spaces