Hardy Space Estimates for Bilinear Square Functions and Calderón-zygmund Operators

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 p1 ×H p2 into Lp for indices smaller than 1, and sufficient BMO type conditions are given for a bilinear Calderón-Zygmund operator to be bounded from H p1 ×H p2 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 p1 ×H p2 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].