Littlewood-Paley-Stein theory and Banach spaces in the inverse Gaussian setting

In this paper we consider Littlewood-Paley functions defined by the semigroups associated with the operator $\mathcal{A}=-\frac{\mathrm{\Delta }}{2}-x\mathrm{\nabla }$ 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 K"othe function spaces with the UMD property.