Lp-regularity of rough bi-parameter Fourier integral operators

Let T be a bi-parameter Fourier integral operator introduced by Hong et al. (De Gruyter 30(1):87–107, 2018) and defined by amplitude a ( x , ξ , η ) ∈ L ∞ B S ϱ m with m = ( m 1 , m 2 ) ∈ R 2 , ϱ = ( ϱ 1 , ϱ 2 ) ∈ [ 0 , 1 ] × [ 0 , 1 ] and phase function φ ( x , ξ , η ) = φ 1 ( x , ξ ) + φ 2 ( x , η ) with φ 1 , φ 2 ∈ L ∞ Φ 2 satisfying the rough non-degeneracy condition, where x = ( x 1 , x 2 ) ∈ R n × R n and η , ξ ∈ R n \ { 0 } . It is showed that T is bounded on L p ( R n ) ( 1 ≤ p ≤ ∞ ) provided that m i < - [ n - 1 2 ϱ i + n 2 ( 1 - ϱ i ) ] ( | 1 2 - 1 p | + 1 2 ) - n 2 ( 1 - ϱ i ) 1 p , i = 1 , 2 .