On the boundedness of rough bi-parameter Fourier integral operators

Let the bi-parameter Fourier integral operators be defined by the phase functions phi(1)(x(1), xi), phi(2)(x(2), xi) is an element of L-infinity Phi(2) satisfying the rough non-degeneracy condition and the amplitude a is an element of(LBS rho m)-B-p with m = (m(1), m(2)) is an element of R-2, rho = (rho(1), rho(2)) is an element of [0, 1] x [0, 1]. It is proved that if 0 < r <= infinity, 1 <= p, q <= 8, satisfying the relation 1/r = 1/q + 1/p, then these operators are bounded from L-q to L-r provided m(i) < - rho(i) (n - 1)/2 (1/s + 1 1/min(p, s') + n(rho(i) - 1)/s i = 1, 2, where s = min(2, p, q) and 1/s + 1/s' = 1.