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

Abstract

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 ℒ r,ϕp/r,κ (μ) into space ℒ p,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 ∈ R B M O ˜ ( μ ) b \in \widetilde {RBMO}\left( \mu \right) and the M ˜ \tilde M α,l,ρq,b on space ℒp (μ) and on space ℒ p , ϕ , κ (μ) is also obtained.