度量测度空间上θ型Marcinkiewicz积分及交换子

数学物理学报 ›› 2020, Vol. 40 ›› Issue (6): 1431-1445.

度量测度空间上θ型Marcinkiewicz积分及交换子

逯光辉*(),陶双平()

西北师范大学数学与统计学院兰州 730070

收稿日期:2020-01-30 出版日期:2020-12-26 发布日期:2020-12-29
通讯作者: 逯光辉 E-mail:luguanghui@nwnu.edu.cn;taosp@nwnu.edu.cn
作者简介:陶双平, E-mail:taosp@nwnu.edu.cn
基金资助:
国家自然科学基金(11561062);甘肃省高等学校科研项目(2020A-010);西北师范大学青年教师科研能力项目(NWNU-LKQN2020-07);西北师范大学博士启动基金(0002020203)

θ Type Marcinkiewicz Integral and Its Commutator on Metric Measure Spaces

Guanghui Lu*(),Shuangping Tao()

College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070

Received:2020-01-30 Online:2020-12-26 Published:2020-12-29
Contact: Guanghui Lu E-mail:luguanghui@nwnu.edu.cn;taosp@nwnu.edu.cn
Supported by:
the NSFC(11561062);the College Scientific Research Project for Colleges of Gansu Province(2020A-010);the Young Teachers Research Ability Project of Northwest Normal University(NWNU-LKQN2020-07);the Scientific Startup Foundation for Doctors of Northwest Normal University(0002020203)

摘要/Abstract

摘要：

设$({\cal X}, d, \mu)$是Hytönen意义下的非齐度量测度空间且满足所谓的几何双倍和上部双倍条件.在假设控制函数$\lambda$满足$\epsilon$-弱逆双倍条件下, 该文主要证明了$\theta$型Marcinkiewicz积分${\cal M}_{\theta}$及由具有离散系数的平均振荡空间$\widetilde{{\rm RBMO}}(\mu)$与${\cal M}_{\theta}$生成的交换子${\cal M}_{\theta, b}$在齐次Herz空间$\dot{K}^{\tau, p}_{q}(\mu)$上的有界性, 也得到了${\cal M}_{\theta}$与${\cal M}_{\theta, b}$从$\widetilde{H}\dot{K}^{\tau, p}_{{\rm atb}, q}(\mu)$到$\dot{K}^{\tau, p}_{q}(\mu)$上的有界性.

关键词: 非齐度量测度空间, θ型Marcinkiewicz积分, 交换子, Lipschitz函数, Herz空间, 原子Herz-Hardy空间

Abstract:

Let $({\cal X}, d, \mu)$ be a non-homogeneous metric measure space satisfying the so-called geometrically doubling and the upper doubling conditions in the sense of ${\rm Hyt\ddot{o}nen}$. Under the assumption that the dominating function $\lambda$ satisfies the $\epsilon$-weak reverse doubling condition, the authors prove that the $\theta$ type Marcinkiewicz integral ${\cal M}_{\theta}$ and the commutator ${\cal M}_{\theta, b}$ generated by the $b\in\widetilde{{\rm RBMO}}(\mu)$ and the ${\cal M}_{\theta}$ is bounded on homogeneous Herz space $\dot{K}^{\tau, p}_{q}(\mu)$, respectively. Furthermore, the boundeness of the ${\cal M}_{\theta}$ and ${\cal M}_{\theta, b}$ from the $\widetilde{H}\dot{K}^{\tau, p}_{{\rm atb}, q}(\mu)$ into the $\dot{K}^{\tau, p}_{q}(\mu)$ is also obtained.

Key words: Non-homogeneous metric measure space, θ type Marcinkiewicz integral, Commutator, Lipschitz function, Herz space, Atomic Herz-Hardy space

中图分类号:

O174.2

逯光辉,陶双平. 度量测度空间上θ型Marcinkiewicz积分及交换子[J]. 数学物理学报, 2020, 40(6): 1431-1445.

Guanghui Lu,Shuangping Tao. θ Type Marcinkiewicz Integral and Its Commutator on Metric Measure Spaces[J]. Acta mathematica scientia,Series A, 2020, 40(6): 1431-1445.

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