Fractional type Marcinkiewicz integrals over non-homogeneous metric measure spaces

*Correspondence: [email protected] College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, People’s Republic of China Abstract The main goal of the paper is to establish the boundedness of the fractional type Marcinkiewicz integralMβ ,ρ ,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 thatMβ ,ρ ,q is bounded from Lebesgue space L1(μ) into the weak Lebesgue space L1,∞(μ), from the Lebesgue space L∞(μ) into the space RBLO(μ), and from the atomic Hardy space H1(μ) into the Lebesgue space L1(μ). Moreover, the authors also get a corollary, that is,Mβ ,ρ ,q is bounded on Lp(μ) with 1 < p <∞. MSC: non-homogeneous metric measure space; fractional type Marcinkiewicz integral; Lebesgue space; Hardy space; RBLO(μ)