Fractional type Marcinkiewicz integrals over non-homogeneous metric measure spaces

Abstract

Abstract The main goal of the paper is to establish the boundedness of the fractional type Marcinkiewicz integral M β , ρ , q $\mathcal{M}_{\beta,\rho,q}$ 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 $\mathcal{M}_{\beta,\rho,q}$ 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 , ρ , q $\mathcal{M}_{\beta,\rho,q}$ is bounded on L p ( μ ) $L^{p}(\mu)$ with 1 < p < ∞ $1< p<\infty$ .

Keywords

• 42B35
• 47B47
• 30L99