Riesz transforms for the Dunkl harmonic oscillator

Abstract

We propose an approach to the theory of Riesz transforms in a framework emerging from certain reflection symmetries in Euclidean spaces. Relying on Rösler’s construction of multivariable generalized Hermite functions associated with a finite reflection group on \({\mathbb R^d}\), we define and investigate a system of Riesz transforms related to the Dunkl harmonic oscillator. In the case isomorphic with the group \({\mathbb{Z}^d_2}\) it is proved that the Riesz transforms are Calderón–Zygmund operators in the sense of the associated space of homogeneous type, thus their mapping properties follow from the general theory.