Littlewood-Paley theory for functions with values in uniformly convex spaces

The aim of this paper is to study the one-sided Littlewood-Paley theory for functions with values in Banach spaces. We characterize, by geometrical properties of the underlying Banach space, the validity in the vector-valued setting of each one-sided inequality contained in the classical L_p-norm inequalities for the Littlewood-Paley g-function and Lusin area integral. The relevant geometrical properties are the uniform convexity and smoothness. When one considers only analytic functions, the uniform convexity is weakened to the uniform PL-convexity and Hardy convexity.