Endpoint estimates for harmonic analysis operators associated with Laguerre polynomial expansions

TL;DR

This paper establishes endpoint boundedness criteria for harmonic analysis operators related to Laguerre polynomial expansions, covering Riesz transforms, maximal functions, and other operators in weighted spaces.

Contribution

It provides a unified criterion for endpoint estimates of various harmonic analysis operators in the Laguerre setting, extending previous results.

Findings

Proved boundedness from Hardy space to L^1 for several operators.

Established boundedness from L^ty to BMO for these operators.

Applied criteria to Riesz transforms, maximal operators, and more.

Abstract

In this paper we give a criterion to prove boundedness results for several operators from $H^1((0,\infty),\gamma_\alpha)$ to $L^1((0,\infty),\gamma_\alpha)$ and also from $L^\infty((0,\infty),\gamma_\alpha)$ to $\BMO((0,\infty),\gamma_\alpha)$, with respect to the probability measure $d\gamma_\alpha (x)=\frac{2}{\Gamma(\alpha+1)} x^{2\alpha+1} e^{-x^2} dx$ on $(0,\infty)$ when ${\alpha>-\frac12}$. We shall apply it to establish endpoint estimates for Riesz transforms, maximal operators, Littlewood-Paley functions, multipliers of Laplace transform type, fractional integrals and variation operators in the Laguerre setting.