A transference result of the $L^p$ continuity of the Jacobi Riesz transform to the Gaussian and Laguerre Riesz transforms

TL;DR

This paper introduces a transference method that leverages asymptotic relations between orthogonal polynomials to transfer $L^p$-continuity results of the Jacobi Riesz transform to the Gaussian and Laguerre Riesz transforms in one dimension.

Contribution

It presents a novel transference technique connecting Jacobi, Gaussian, and Laguerre Riesz transforms via polynomial asymptotics.

Findings

Established $L^p$-continuity for Gaussian Riesz transform from Jacobi case.

Extended $L^p$-continuity to Laguerre Riesz transform using the same approach.

Demonstrated the effectiveness of asymptotic relations in harmonic analysis transference.

Abstract

In this paper using the well known asymptotic relations between Jacobi polynomials and Hermite and Laguerre polynomials. We develop a transference method to obtain the $L^p$-continuity of the Gaussian-Riesz transform and the $L^p$-continuity of the Laguerre-Riesz transform from the $L^p$-continuity of the Jacobi-Riesz transform, in dimension one.