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

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.

References in corpus (1)

• $\gamma$-Radonifying operators -- a survey

Cited by in corpus (7)

• The fractional Bessel equation in Hölder spaces
• UMD Banach spaces and square functions associated with heat semigroups for Schrödinger and Laguerre operators
• The Hardy-Littlewood property and maximal operators associated with the inverse Gauss measure
• Conical square functions associated with Bessel, Laguerre and Schrödinger operators in UMD Banach spaces
• UMD-valued square functions associated with Bessel operators in Hardy and BMO spaces
• Discrete harmonic analysis associated with ultraspherical expansions
• Variation operators for semigroups associated with Fourier-Bessel expansions