Littlewood-Paley-Stein Theory and Banach Spaces in the Inverse Gaussian Setting

Abstract

In this paper we consider Littlewood-Paley functions defined by the semigroups associated with the operator mathcal {A}=-frac {1}{2}{Delta }-xnabla in the inverse Gaussian setting for Banach valued functions. We characterize the uniformly convex and smooth Banach spaces by using L^{p}(mathbb R^{n},gamma _{-1})- properties of the mathcal {A}-Littlewood-Paley functions. We also use Littlewood-Paley functions associated with mathcal {A} to characterize the Köthe function spaces with the UMD property.