Calder\'on-Zygmund operators related to Laguerre function expansions of convolution type

TL;DR

This paper introduces a new technique for proving standard estimates for Laguerre function expansions of convolution type, applicable to all admissible multi-indices, and demonstrates that key harmonic analysis operators are Calderón-Zygmund operators in this setting.

Contribution

It generalizes existing methods to all admissible multi-indices and establishes that fundamental harmonic analysis operators are Calderón-Zygmund operators in the Laguerre setting.

Findings

Proved maximal operators related to heat and Poisson semigroups are Calderón-Zygmund operators.

Established Riesz transforms and square functions as Calderón-Zygmund operators.

Extended the class of operators known to be Calderón-Zygmund in Laguerre analysis.

Abstract

We develop a technique of proving standard estimates in the setting of Laguerre function expansions of convolution type, which works for all admissible type multi-indices $\alpha$ in this context. This generalizes a simpler method existing in the literature, but being valid for a restricted range of $\alpha$. As an application, we prove that several fundamental operators in harmonic analysis of the Laguerre expansions, including maximal operators related to the heat and Poisson semigroups, Riesz transforms, Littlewood-Paley-Stein type square functions and multipliers of Laplace and Laplace-Stieltjes transforms type, are (vector-valued) Calder\'on-Zygmund operators in the sense of the associated space of homogeneous type.