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