We prove the boundedness of a class of tri-linear operators consisting of a quasi piece of bilinear Hilbert transform whose scale equals to or dominates the scale of its linear counter part. Such type of operators is motivated by the tri-linear Hilbert transform and its curved versions.
We introduce a class of tri-linear operators that combine features of the bilinear Hilbert transform and paraproduct. For two instances of these operators, we prove boundedness property in a large range D = { ( p 1 , p 2 , p 3 ) 3 : 1 < p 1