Imaginary powers of $(k,1)$-generalized harmonic oscillator

Abstract

In this paper we will define and investigate the imaginary powers ${\left(-{△}_{k,1}\right)}^{-i\sigma },\sigma \in \mathbb{R}$ of the $(k,1)$-generalized harmonic oscillator $-{△}_{k,1}=-\Vert x\Vert {△}_k+\Vert x\Vert$ and prove the $L^p$-boundedness $(1<p<\infty )$ and weak $L^1$-boundedness of such operators. It is a parallel result to the $L^p$-boundedness $(1<p<\infty )$ and weak $L^1$-boundedness of the imaginary powers of the Dunkl harmonic oscillator $-{△}_k+{\Vert x\Vert }^2$. To prove this result, we develop the Calderón--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 ${\left(-{△}_{k,1}\right)}^{-i\sigma }$ are singular integral operators satisfying the corresponding Hörmander type condition.

accepted for publication in Complex Analysis and Operator Theory

Cited by in corpus (1)

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