The Riesz transform for the harmonic oscillator in spherical coordinates

TL;DR

This paper establishes weighted mixed norm estimates for the Riesz transform linked to the harmonic oscillator in spherical coordinates, utilizing Laguerre expansions and advanced extrapolation techniques.

Contribution

It introduces new weighted inequalities for Laguerre-related Riesz transforms and adapts Rubio de Francia's extrapolation to this context.

Findings

Weighted inequalities for Laguerre Riesz transforms established

Extension of Rubio de Francia's extrapolation theorem to this setting

Mixed norm estimates for the harmonic oscillator Riesz transform proved

Abstract

In this paper we show weighted estimates in mixed norm spaces for the Riesz transform associated with the harmonic oscillator in spherical coordinates. In order to prove the result we need a weighted inequality for a vector-valued extension of the Riesz transform related to the Laguerre expansions which is of independent interest. The main tools to obtain such extension are a weighted inequality for the Riesz transform independent of the order of the involved Laguerre functions and an appropriate adaptation of Rubio de Francia's extrapolation theorem.