Riesz transforms, Hardy spaces and Campanato spaces associated with Laguerre expansions

TL;DR

This paper proves the Riesz transform associated with Laguerre expansions is a Calderón-Zygmund operator, and develops Hardy and Campanato spaces for these operators, establishing their boundedness and advancing harmonic analysis in this setting.

Contribution

It establishes the Calderón-Zygmund nature of the Riesz transform for Laguerre operators and develops associated Hardy and Campanato spaces, filling a gap in the harmonic analysis theory.

Findings

Riesz transform is a Calderón-Zygmund operator

Boundedness of Riesz transform on Hardy and Campanato spaces

Completes the harmonic analysis framework for Laguerre expansions

Abstract

Let $\nu\in [-1/2,\infty)^n$, $n\ge 1$, and let $\mathcal{L}_\nu$ be a self-adjoint extension of the differential operator \[ L_\nu := \sum_{i=1}^n \left[-\frac{\partial^2}{\partial x_i^2} + x_i^2 + \frac{1}{x_i^2}(\nu_i^2 - \frac{1}{4})\right] \] on $C_c^\infty(\mathbb{R}_+^n)$ as the natural domain. In this paper, we first prove that the Riesz transform associated with $\mathcal L_\nu$ is a Calder\'on-Zygmund operator, answering the open problem in [JFA, 244 (2007), 399-443]. In addition, we develop the theory of Hardy spaces and Campanato spaces associated with $\mathcal{L}_\nu$. As applications, we prove that the Riesz transform related to $\mathcal{L}_\nu$ is bounded on these Hardy spaces and Campanato spaces, completing the description of the boundedness of the Riesz transform in the Laguerre expansion setting.