LAUSR

Abstract Fourier integral operators play an important role in applications to partial differential equations. The main purpose of this paper is to investigate the weighted L p boundedness of the bi-parameter Fourier integral operators (or FIOs for short) of the following form: T ( f ) ( x ) = ∫ R n 1 × R n 2 e i φ ( x , ξ , η ) ⋅ a ( x , ξ , η ) ⋅ f ˆ ( ξ , η ) d ξ d η , where for x = ( x 1 , x 2 ) ∈ R n 1 × R n 2 , ξ ∈ R n 1 ∖ { 0 } and η ∈ R n 2 ∖ { 0 } , the amplitude function a ( x , ξ , η ) ∈ L ∞ B S ρ m and the phase function is of the form φ ( x , ξ , η ) = φ 1 ( x 1 , ξ ) + φ 2 ( x 2 , η ) , when φ 1 ∈ L ∞ Φ 2 ( R n 1 × R n 1 ∖ { 0 } ) , φ 2 ∈ L ∞ Φ 2 ( R n 2 × R n 2 ∖ { 0 } ) satisfy certain differential conditions. We will establish the weighted L w p boundedness of T under the assumption that the weight w is in the class of the product Muckenhoupt A p ( R n 1 × R n 2 ) weights.