Imaginary Powers of (K,1)-generalized Harmonic Oscillator

Abstract See Manuscript PDF file for complete Abstract.In this paper we will define and investigate the imaginary powers of the (k,1)-generalized harmonic oscillator −4 k, 1 = −kxk4 k + kxk and prove the Lp-boundedness (1 < p < ∞) and weak L1-boundedness of such operators. It is a parallel result to the Lp-boundedness (1 < p < ∞) and weak L1-boundedness of the imaginary powers of the Dunkl harmonic oscillator −4 k + kxk2. To prove this result, we develop the Calder´on–Zygmund theory adapted to the (k,1)-generalized setting by constructing the metric space of homogeneous type corresponding to the (k,1)-generalized setting, and show that are singular integral operators satisfying the corresponding H¨ormander type condition.