Fractional Type Marcinkiewicz Commutators Over Non-Homogeneous Metric Measure Spaces

Abstract

The main purpose of this paper is to establish the boundedness of the commutator $$\mathcal{M}_{\beta,\rho,m,b}$$ generated by the fractional type Marcinkiewicz integral $$\mathcal{M}_{\beta,\rho,m}$$ with Lipschitz function b on non-homogeneous metric measure spaces, which satisfy the geometrically doubling condition and the upper doubling condition. Under the assumption that the dominating function λ satisfies the e-weak reverse doubling condition, the authors prove that $$\mathcal{M}_{\beta,\rho,m,b}$$ is bounded from the Lebesgue space L1(μ) to the Lebesgue space $$L^{\frac{1}{1-\alpha},\infty}(\mu)$$ , from the Lebesgue space Lp(μ) into the Lebesgue space Lq(μ), from the Lebesgue space Lp(μ) to the Lipschitz space $${\rm{Lip}}_{(\alpha-\frac{1}{p})}(\mu)$$ , and from the Hardy space H1(μ) into the Lebesgue space Lq(μ). Moreover, the authors also prove that $$\mathcal{M}_{\beta,\rho,m,b}$$ is bounded from the Lebesgue space Lp(μ) into the space RBMO(μ) when $$p = \frac{1}{\alpha}$$ .