$\gamma$-radonifying operators and UMD-valued Littlewood-Paley-Stein functions in the Hermite setting on BMO and Hardy spaces

TL;DR

This paper investigates Hermite Littlewood-Paley-Stein functions for functions valued in UMD Banach spaces, establishing boundedness properties and new characterizations of UMD spaces within the Hermite setting on BMO and Hardy spaces.

Contribution

It introduces boundedness results for Hermite square functions on vector-valued BMO and Hardy spaces, providing new characterizations of UMD Banach spaces.

Findings

Hermite square functions are bounded from BMO_L(R, B) to BMO_L(R, γ(H, B))

Equivalent norms in BMO_L and H^1_L spaces via Littlewood-Paley-Stein functions

New characterizations of UMD Banach spaces based on Hermite Littlewood-Paley theory

Abstract

In this paper we study Littlewood-Paley-Stein functions associated with the Poisson semigroup for the Hermite operator on functions with values in a UMD Banach space $\B.$ If we denote by $H$ the Hilbert space $L^2((0,\infty),dt/t),\gamma(H,\B)$ represents the space of $\gamma$-radonifying operators from $H$ into $\B.$ We prove that the Hermite square function defines bounded operators from $BMO_\mathcal{L}(\R,\B)$ (respectively, $H^1_\mathcal{L}(\R, \B)$) into $BMO_\mathcal{L}(\R,\gamma(H,\B))$ (respectively, $H^1_\mathcal{L}(\R, \gamma(H,\B))$), where $BMO_\mathcal{L}$ and $H^1_\mathcal{L}$ denote $BMO$ and Hardy spaces in the Hermite setting. Also, we obtain equivalent norms in $BMO_\mathcal{L}(\R, \B)$ and $H^1_\mathcal{L}(\R,\B)$ by using Littlewood-Paley-Stein functions. As a consequence of our results, we establish new characterizations of the UMD Banach spaces.