Imaginary Powers of the Dunkl Harmonic Oscillator

Abstract

In this paper we continue the study of spectral properties of the Dunkl harmonic oscillator in the context of a finite reflection group on ${\mathbb{R}}^d$ isomorphic to ${\mathbb{Z}}_2^d$. We prove that imaginary powers of this operator are bounded on $L^p$, $1<p<\infty$, and from $L^1$ into weak $L^1$.