Conical square functions associated with Bessel, Laguerre and Schr\"odinger operators in UMD Banach spaces

TL;DR

This paper develops Banach space-valued conical square functions associated with Bessel, Laguerre, and Schrödinger operators, providing new equivalent norms in Lebesgue-Bochner spaces using $\,\gamma$-radonifying operators.

Contribution

It introduces Banach space-valued conical square functions in these settings and establishes new equivalent norms in Lebesgue-Bochner spaces, extending scalar results to UMD Banach spaces.

Findings

New equivalent norms in Lebesgue-Bochner spaces using square functions.

Extension of scalar square function results to Banach space-valued functions.

Applicability to UMD Banach spaces for Bessel, Laguerre, and Schrödinger operators.

Abstract

In this paper we consider conical square functions in the Bessel, Laguerre and Schr\"odinger settings where the functions take values in UMD Banach spaces. Following a recent paper of Hyt\"onen, van Neerven and Portal, in order to define our conical square functions, we use $\gamma$-radonifying operators. We obtain new equivalent norms in the Lebesgue-Bochner spaces $L^p((0,\infty ),\mathbb{B})$ and $L^p(\mathbb{R}^n,\mathbb{B})$, $1<p<\infty$, in terms of our square functions, provided that $\mathbb{B}$ is a UMD Banach space. Our results can be seen as Banach valued versions of known scalar results for square functions.