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\varphi(x,\xi,\eta)}%
\cdot a(x,\xi,\eta)\cdot\widehat{f}(\xi,\eta)\,d\xi\,d\eta,
where
{x=(x_{1},x_{2})\in\mathbb{R}^{n}\times\mathbb{R}^{n}}
and
{\xi,\eta\in\mathbb{R}^{n}\setminus\{0\}}
,
{a(x,\xi,\eta)\in L^{\infty}BS^{m}_{\rho}}
is the amplitude,
and the phase function is of the form
\varphi(x,\xi,\eta)=\varphi_{1}(x_{1},\xi\/)+\varphi_{2}(x_{2},\eta)
, with
\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}}
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}BS^{m}_{\rho}}
and non-smooth phase function
{\varphi(x,\xi,\eta)}
on x satisfying a rough non-degeneracy condition.
Trace ideals for pseudo-differential operators and their commutators with symbols in α-modulation spaces
