Higher Order Riesz Transforms for Laguerre Expansions

TL;DR

This paper studies the boundedness and singular integral properties of higher order Riesz transforms linked to Laguerre operators, using a novel identity connecting Hermite and Laguerre frameworks.

Contribution

It introduces a new identity connecting Hermite and Laguerre Riesz transforms and establishes boundedness and principal value properties for these transforms.

Findings

Lp-boundedness of higher order Riesz transforms

Riesz transforms are principal value singular integrals

New identity linking Hermite and Laguerre Riesz transforms

Abstract

In this paper we investigate Lp-boundedness properties for the higher order Riesz transforms associated with Laguerre operators. Also we prove that the k-th Riesz transform is a principal value singular integral operator (modulus a constant times of the function when k is even). To establish our results we exploit a new identity connecting Riesz transforms in the Hermite and Laguerre settings.