Fractional Type Marcinkiewicz Integral Operator Associated with Θ-Type Generalized Fractional Kernel and Its Commutator on Non-homogeneous Spaces

Abstract Let (𝒳, d, μ) be a non-homogeneous metric measure space satisfying the upper doubling and geometrically doubling conditions in the sense of Hytönen. Under assumption that θ and dominating function λ satisfy certain conditions, the authors prove that fractional type Marcinkiewicz integral operator M˜ \tilde M α,lρ,q associated with θ-type generalized fractional kernel is bounded from the generalized Morrey space ℒr,ϕp/r,κ (μ) into space ℒp,ϕ,κ (μ), and bounded from the Lebesgue space Lr(μ) into space Lp(μ). Furthermore, the boundedness of commutator M˜ \tilde M α,l,ρq,b generated by b∈RBMO˜(μ) b \in \widetilde {RBMO}\left( \mu \right) and the M˜ \tilde M α,l,ρq,b on space ℒp(μ) and on space ℒp,ϕ,κ (μ) is also obtained.