Fractional type Marcinkiewicz integrals over non-homogeneous metric measure spaces

摘要

The main goal of the paper is to establish the boundedness of the fractional type Marcinkiewicz integral M β , ρ , q $mathcal{M}_{eta,ho,q}$ on non-homogeneous metric measure space which includes the upper doubling and the geometrically doubling conditions. Under the assumption that the kernel satisfies a certain Hörmander-type condition, the authors prove that M β , ρ , q $mathcal{M}_{eta,ho,q}$ is bounded from Lebesgue space L 1 ( μ ) $L^{1}(mu)$ into the weak Lebesgue space L 1 , ∞ ( μ ) $L^{1,infty}(mu)$ , from the Lebesgue space L ∞ ( μ ) $L^{infty}(mu)$ into the space RBLO ( μ ) $operatorname{RBLO}(mu)$ , and from the atomic Hardy space H 1 ( μ ) $H^{1}(mu)$ into the Lebesgue space L 1 ( μ ) $L^{1}(mu)$ . Moreover, the authors also get a corollary, that is, M β ,