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

TL;DR

This paper defines and studies the imaginary powers of a generalized harmonic oscillator, proving their boundedness on L^p spaces by developing a specialized Calderón–Zygmund theory for this setting.

Contribution

It introduces a new framework for analyzing the imaginary powers of the $(k,1)$-generalized harmonic oscillator, extending Calderón–Zygmund theory to this context.

Findings

Proved L^p-boundedness for the imaginary powers of the generalized harmonic oscillator.

Established weak L^1-boundedness of these operators.

Developed a Calderón–Zygmund theory adapted to the generalized setting.

Abstract

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.