Lp 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) = 1/(2 pi)(2n) integral(R2n) e(i phi(x, xi, eta)) center dot a(x, xi, eta) center dot(f) over cap(xi, eta) d xi d eta, where x = (x(1), x(2)) epsilon R-n x R-n and xi, eta epsilon R-n\{0}, a(x, xi, eta) epsilon (LBS rho m)-B-infinity is the amplitude, 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) epsilon L-infinity Phi(2) (R-n x R-n\{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) epsilon (LBS rho m)-B-infinity and non-smooth phase function phi(x, xi, eta) on x satisfying a rough non-degeneracy condition.