L p boundedness of rough bi-parameter Fourier integral operators

Abstract

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 ) = 1 ( 2 ⁢ π ) 2 ⁢ n ⁢ ∫ ℝ 2 ⁢ n e i ⁢ φ ⁢ ( x , ξ , η ) ⋅ a ⁢ ( x , ξ , η ) ⋅ f ^ ⁢ ( ξ , η ) ⁢ 𝑑 ξ ⁢ 𝑑 η , T(f\/)(x)=\frac{1}{(2\pi)^{2n}}\int_{\mathbb{R}^{2n}}e^{i\varphi(x,\xi,\eta)}% \cdot a(x,\xi,\eta)\cdot\widehat{f}(\xi,\eta)\,d\xi\,d\eta, where x = ( x 1 , x 2 ) ∈ ℝ n × ℝ n {x=(x_{1},x_{2})\in\mathbb{R}^{n}\times\mathbb{R}^{n}} and ξ , η ∈ ℝ n ∖ { 0 } {\xi,\eta\in\mathbb{R}^{n}\setminus\{0\}} , a ⁢ ( x , ξ , η ) ∈ L ∞ ⁢ B ⁢ S ρ m {a(x,\xi,\eta)\in L^{\infty}BS^{m}_{\rho}} is the amplitude, and the phase function is of the form φ ⁢ ( x , ξ , η ) = φ 1 ⁢ ( x 1 , ξ ) + φ 2 ⁢ ( x 2 , η ) \varphi(x,\xi,\eta)=\varphi_{1}(x_{1},\xi\/)+\varphi_{2}(x_{2},\eta) , with φ 1 , φ 2 ∈ L ∞ ⁢ Φ 2 ⁢ ( ℝ n × ℝ n ∖ { 0 } ) \varphi_{1},\varphi_{2}\in L^{\infty}\Phi^{2}(\mathbb{R}^{n}\times\mathbb{R}^{% n}\setminus\{0\}) , and satisfies a certain rough non-degeneracy condition (see (2.2)). The study of these operators are motivated by the L p {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 {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 {L^{p}} boundedness of the more general FIOs with amplitude a ⁢ ( x , ξ , η ) ∈ L ∞ ⁢ B ⁢ S ρ m {a(x,\xi,\eta)\in L^{\infty}BS^{m}_{\rho}} and non-smooth phase function φ ⁢ ( x , ξ , η ) {\varphi(x,\xi,\eta)} on x satisfying a rough non-degeneracy condition.