On a hybrid of bilinear Hilbert transform and paraproduct

Abstract

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 \(D = \left\{ {\left( {{p_1},{p_2},{p_3}} \right) \in {\mathbb{R}^3}:1 < {p_1},{p_2} < \infty ,\frac{1}{{{p_1}}} + \frac{1}{{{p_2}}} < \frac{3}{2},1 < {p_3} < \infty } \right\}\).