Hardy space estimates for bilinear square functions and Calderon-Zygmund operators

Abstract

In this work we prove Hardy space estimates for bilinear Littlewood-Paley-Stein square function and Calderón-Zygmund operators. Sufficient Carleson measure type conditions are given for square functions to be bounded from H p 1 × H p 2 into L p for indices smaller than 1, and sufficient BMO type conditions are given for a bilinear Calderón-Zygmund operator to be bounded from H p 1 × H p 2 into H p for indices smaller than 1. Subtle difficulties arise in the bilinear nature of these problems that are related to frequency properties of products of functions. Moreover, three types of bilinear paraproducts are defined and shown to be bounded from H p 1 × H p 2 into H p for indices smaller than 1. The first is a bilinear Bony type paraproduct that was defined in [33]. The second is a paraproduct that resembles the product of two Hardy space functions. The third class of paraproducts are operators given by sums of molecules, which were introduced in [2].

Conditions for boundedness into Hardy spaces

“…Our approach is powerful enough to encompass many types of multilinear operators that include all the previously studied (Coifman-Meyer type and finite sums of products of Calderón-Zygmund operators), as well as mixed types. An alternative approach to Hardy space estimates for bilinear operators has appeared in the recent work of Hart and Lu [12].Recall that the Hardy space with 0 < < ∞ is given as the space of all tempered distributions for which…”