On boundedness of discrete multilinear singular integral operators

Abstract

Let m ( ξ , η ) be a measurable locally bounded function defined in R 2 . Let 1 ⩽ p 1 , q 1 , p 2 , q 2 < ∞ such that p i = 1 implies q i = ∞ . Let also 0 < p 3 , q 3 < ∞ and 1 / p = 1 / p 1 + 1 / p 2 − 1 / p 3 . We prove the following transference result: the operator C m ( f , g ) ( x ) = ∫ R ∫ R f ˆ ( ξ ) g ˆ ( η ) m ( ξ , η ) e 2 π i x ( ξ + η ) d ξ d η initially defined for integrable functions with compact Fourier support, extends to a bounded bilinear operator from L p 1 , q 1 ( R ) × L p 2 , q 2 ( R ) into L p 3 , q 3 ( R ) if and only if the family of operators D m ˜ t , p ( a , b ) ( n ) = t 1 p ∫ − 1 2 1 2 ∫ − 1 2 1 2 P ( ξ ) Q ( η ) m ( t ξ , t η ) e 2 π i n ( ξ + η ) d ξ d η initially defined for finite sequences a = ( a k 1 ) k 1 ∈ Z , b = ( b k 2 ) k 2 ∈ Z , where P ( ξ ) = ∑ k 1 ∈ Z a k 1 e − 2 π i k 1 ξ and Q ( η ) = ∑ k 2 ∈ Z b k 2 e − 2 π i k 2 η , extend to bounded bilinear operators from l p 1 , q 1 ( Z ) × l p 2 , q 2 ( Z ) into l p 3 , q 3 ( Z ) with norm bounded by uniform constant for all t > 0 . We apply this result to prove boundedness of the discrete Bilinear Hilbert transforms and other related discrete multilinear singular integrals including the endpoints.