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

Abstract

In this paper, we will investigate the boundedness of the bi-parameter Fourier integral operators (or FIOs for short) of the following form: $$T(f)(x)=\frac{1}{(2\pi {)}^{2n}}{\int }_{{\mathbb{R}}^{2n}}{e}^{i\phi (x,\xi ,\eta )}\cdot a(x,\xi ,\eta )\cdot \hat{f}(\xi ,\eta )d\xi d\eta ,$$ where for $x=(x_1,x_2)\in {\mathbb{R}}^n\times {\mathbb{R}}^n$ and $\xi ,\eta \in {\mathbb{R}}^n\setminus \{0\}$, the amplitude $a(x,\xi ,\eta )\in L^{\infty }B{S}_{\rho }^m$ and the phase function is of the form $\phi (x,\xi ,\eta )={\phi }_1(x_1,\xi )+{\phi }_2(x_2,\eta )$ with ${\phi }_1,{\phi }_2\in L^{\infty }{\Phi }^2({\mathbb{R}}^n\times {\mathbb{R}}^n\setminus \{0\})$ and $\phi (x,\xi ,\eta )$ 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 $a(x,\xi ,\eta )\in L^{\infty }B{S}_{\rho }^m$ and non-smooth phase function $\phi (x,\xi ,\eta )$ on $x$ satisfying a rough non-degeneracy condition.

29 pages