Acta mathematica scientia,Series A ›› 2020, Vol. 40 ›› Issue (6): 1431-1445.

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θ Type Marcinkiewicz Integral and Its Commutator on Metric Measure Spaces

Guanghui Lu*(),Shuangping Tao()   

  1. 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;
  • 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)


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

CLC Number: 

  • O174.2