Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 016, 12 pages      arXiv:0902.1958      doi:10.3842/SIGMA.2009.016
Contribution to the Special Issue on Dunkl Operators and Related Topics

Imaginary Powers of the Dunkl Harmonic Oscillator

Adam Nowak and Krzysztof Stempak
Instytut Matematyki i Informatyki, Politechnika Wroclawska, Wyb. Wyspianskiego 27, 50-370 Wroclaw, Poland

Received October 14, 2008, in final form February 08, 2009; Published online February 11, 2009

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 Rd isomorphic to Z2d. We prove that imaginary powers of this operator are bounded on Lp, 1 < p < ∞, and from L1 into weak L1.

Key words: Dunkl operators; Dunkl harmonic oscillator; imaginary powers; Calderón-Zygmund operators.

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