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

TL;DR

This paper explores Littlewood-Paley theory in the inverse Gaussian setting to characterize Banach space properties and K"othe function spaces with the UMD property using semigroup-based functions.

Contribution

It introduces new characterizations of Banach space convexity, smoothness, and UMD property via Littlewood-Paley functions associated with a specific inverse Gaussian operator.

Findings

Characterizes uniformly convex Banach spaces using Littlewood-Paley functions.

Identifies smooth Banach spaces through $L^p$-properties of Littlewood-Paley functions.

Provides criteria for K"othe function spaces to have the UMD property.

Abstract

In this paper we consider Littlewood-Paley functions defined by the semigroups associated with the operator $\mathcal{A}=-\frac{\Delta}{2}-x\nabla$ 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.