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

Abstract

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