On the boundedness of rough bi-parameter Fourier integral operators

Let the bi-parameter Fourier integral operators be defined by the phase functions φ 1 ( x 1 , ξ ) , φ 2 ( x 2 , ξ ) ∈ L ∞ Φ 2 satisfying the rough non-degeneracy condition and the amplitude a ∈ L p B S ϱ m with m = ( m 1 , m 2 ) ∈ R 2 , ϱ = ( ϱ 1 , ϱ 2 ) ∈ [ 0 , 1 ] × [ 0 , 1 ] . It is proved that if 0 < r ≤ ∞ , 1 ≤ p , q ≤ ∞ , satisfying the relation 1 r = 1 q + 1 p , then these operators are bounded from L q to L r provided m i < - ϱ i ( n - 1 ) 2 ( 1 s + 1 min ( p , s ′ ) ) + n ( ϱ i - 1 ) s i = 1 , 2 , where s = min ( 2 , p , q ) and 1 s + 1 s ′ = 1 .