The Riesz Transform for the Harmonic Oscillator in Spherical Coordinates

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 that is of independent interest. The main tools to obtain such an 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 Francias extrapolation theorem