The purpose of this dissertation is to provide a new square function characterization of weak Hardy spaces in the full range of exponents possible and use this characterization in applications on endpoint estimates for multilinear paraproducts.
Additionally, we prove several maximal characterizations of weak Hardy spaces and obtain several properties of these spaces. Our main result is a Littlewood-Paley square function characterization of the Hardy spaces. Our proof is based on a Calderón-Zygmund type decomposition of distributions in Hardy spaces and on interpolation. Our results allow us to obtain endpoint estimates for several operators in terms of square function characterizations of weak L^p norms. As an application of this technique, we prove endpoint boundedness for mutlilinear paraproducts.