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

In this paper we will define and investigate the imaginary powers $\left(-\triangle_{k,1}\right)^{-i\sigma},\sigma\in\mathbb{R}$ of the $(k,1)$-generalized harmonic oscillator $-\triangle_{k,1}=-\left\|x\right\|\triangle_k+\left\|x\right\|$ 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 $-\triangle_k+\left\|x\right\|^2$. 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 $\left(-\triangle_{k,1}\right)^{-i\sigma}$ are singular integral operators satisfying the corresponding H\"ormander type condition.