Rough bilinear singular integrals

Abstract We study the rough bilinear singular integral, introduced by Coifman and Meyer [8] , T Ω ( f , g ) ( x ) = p.v. ∫ R n ∫ R n | ( y , z ) | − 2 n Ω ( ( y , z ) / | ( y , z ) | ) f ( x − y ) g ( x − z ) d y d z , when Ω is a function in L q ( S 2 n − 1 ) with vanishing integral and 2 ≤ q ≤ ∞ . When q = ∞ we obtain boundedness for T Ω from L p 1 ( R n ) × L p 2 ( R n ) to L p ( R n ) when 1 p 1 , p 2 ∞ and 1 / p = 1 / p 1 + 1 / p 2 . For q = 2 we obtain that T Ω is bounded from L 2 ( R n ) × L 2 ( R n ) to L 1 ( R n ) . For q between 2 and infinity we obtain the analogous boundedness on a set of indices around the point ( 1 / 2 , 1 / 2 , 1 ) . To obtain our results we introduce a new bilinear technique based on tensor-type wavelet decompositions.