Square functions in the Hermite setting for functions with values in UMD spaces

TL;DR

This paper characterizes vector-valued Lebesgue spaces using Hermite Littlewood-Paley functions and identifies UMD Banach spaces through boundedness properties of these functions in the Hermite setting.

Contribution

It introduces a new characterization of UMD Banach spaces via Hermite Littlewood-Paley g-functions in the context of vector-valued Lebesgue spaces.

Findings

Characterization of $L^p( ^n,B)$ spaces using Hermite Littlewood-Paley g-functions.

Identification of UMD Banach spaces through boundedness of Hermite g-functions.

Use of $b3$-radonifying operators in the analysis.

Abstract

In this paper we characterize the Lebesgue Bochner spaces $L^p(\mathbb{R}^n,B)$, $1<p<\infty$, by using Littlewood-Paley $g$-functions in the Hermite setting, provided that $B$ is a UMD Banach space. We use $\gamma$-radonifying operators $\gamma (H,B)$ where $H=L^2((0,\infty),\frac{dt}{t})$. We also characterize the UMD Banach spaces in terms of $L^p(\mathbb{R}^n,B)$-$L^p(\mathbb{R}^n,\gamma (H,B))$ boundedness of Hermite Littlewood-Paley $g$-functions.