$L^p$ Boundedness of rough Bi-parameter Fourier Integral Operators

In this paper, we will investigate the boundedness of the bi-parameter Fourier integral operators (or FIOs for short) of the following form: $$T(f)(x)=\frac{1}{(2\pi {)}^{2n}}{\int }_{{\mathbb{R}}^{2n}}{e}^{i\phi (x,\xi ,\eta )}\cdot a(x,\xi ,\eta )\cdot \widehat{f}(\xi ,\eta )d\xi d\eta ,$$ where for $x=(x_1,x_2)\in $\mathbb{R}$^{n}\times $\mathbb{R}$^{n}$and$\xi,\eta \in $\mathbb{R}$^{n}\setminus{0}$,theamplitude$a(x,\xi,\eta)\in L^\infty BS^m_\rho$andthephasefunctionisoftheform$ \varphi(x,\xi,\eta)=\varphi_1(x_1,\xi)+\varphi_2(x_2,\eta)$with$\quad \varphi_1,\varphi_2 \in L^\infty \Phi^2 ($\mathbb{R}$^{n}\times$\mathbb{R}$^{n}\setminus{0})$and$\varphi(x, \xi, \eta)$ satisfies a certain rough non-degeneracy condition. The study of these operators are motivated by the $L^p$ estimates for one-parameter FIOs and bi-parameter Fourier multipliers and pseudo-differential operators. We will first define the bi-parameter FIOs and then study the $L^p$ boundedness of such operators when their phase functions have compact support in frequency variables with certain necessary non-degeneracy conditions. We will then establish the $L^p$ boundedness of the more general FIOs with amplitude $a(x,\xi ,\eta )\in L^{\mathrm{\infty }}B{S}_{\rho }^m$ and non-smooth phase function $\phi (x,\xi ,\eta )$ on $x$ satisfying a rough non-degeneracy condition.