Weighted norm inequalities of some singular integrals associated with Laguerre expansions

TL;DR

This paper establishes weighted inequalities for singular integrals related to Laguerre expansions, including maximal functions, Riesz transforms, and square functions, extending the understanding of these operators for a full range of parameters.

Contribution

It provides new weighted estimates for singular integrals in the Laguerre setting, especially completing the Riesz transform characterization for all parameter ranges.

Findings

Weighted estimates for maximal functions, Riesz transforms, and square functions in Laguerre setting.

Complete description of Riesz transform for all u in (-1,∞)^n.

Improved results over previous work for certain parameter ranges.

Abstract

Let $\nu=(\nu_1,\ldots,\nu_n)\in (-1,\infty)^n$, $n\ge 1$, 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 investigate the weighted estimates of singular integrals in the Laguerre setting including the maximal function, the Riesz transform and the square functions associated to the Laguerre operator $\mathcal L_\nu$. In the special case of the Riesz transform, the paper completes the description of the Riesz transform for the full range of $\nu\in (-1,\infty)^n$ which significant improves the result in [J. Funct. Anal. 244 (2007), 399--443] for $\nu_i\ge -1/2, \nu_i\notin (-1/2,1/2)$ for $i=1,\ldots, n$.