$(k,1)$ -广义谐振子的虚幂
Imaginary powers of $(k,1)$-generalized harmonic oscillator
摘要：
本文定义并研究了$(k,1)$ -广义谐振子$-\triangle_{k,1}=-\left\|x\right\|\triangle_k+\left\|x\right\|$的虚幂$\left(-\triangle_{k,1}\right)^{-iσ},σ\in\mathbb{R}$，证明了这类算子的$L^p$ -有界性$(1<p<\infty)$和弱$L^1$ -有界性.这是与Dunkl谐振子$-\triangle_k+\left\|x\right\|^2$虚幂的$L^p$ -有界性$(1<p<\infty)$和弱$L^1$ -有界性平行的结果.为了证明这一结果，我们通过构造对应于$(k,1)$-广义集的齐型度量空间，发展了适合于$(k,1)$-广义集的Calderon-Zygmund理论，并证明了$\left(-\triangle_{k,1}\right)^{-iσ}$是满足相应Hörmander型条件的奇异积分算子.
英文摘要：
In this paper we will define and investigate the imaginary powers $\left(-\triangle_{k,1}\right)^{-iσ},σ\in\mathbb{R}$ 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 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 $\left(-\triangle_{k,1}\right)^{-iσ}$ are singular integral operators satisfying the corresponding Hörmander type condition.
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