Marcinkiewicz Integrals with Non-Doubling Measures

Let μ be a positive Radon measure on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${\mathbb{R}}^d$$ \end{document} which may be non doubling. The only condition that μ must satisfy is μ(B(x, r)) ≤ Crn for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$x \in {\mathbb{R}}^d$$ \end{document} , r > 0 and some fixed constants C > 0 and n ∈ (0, d]. In this paper, we introduce the Marcinkiewicz integral related to a such measure with kernel satisfying some Hörmander-type condition, and assume that it is bounded on L2(μ). We then establish its boundedness, respectively, from the Lebesgue space L1(μ) to the weak Lebesgue space L1,∞(μ), from the Hardy space H1(μ) to L1(μ) and from the Lebesgue space L∞(μ) to the space RBLO(μ). As a corollary, we obtain the boundedness of the Marcinkiewicz integral in the Lebesgue space Lp(μ) with p ∈ (1,∞). Moreover, we establish the boundedness of the commutator generated by the RBMO(μ) function and the Marcinkiewicz integral with kernel satisfying certain slightly stronger Hörmander-type condition, respectively, from Lp(μ) with p ∈ (1,∞) to itself, from the space L log L(μ) to L1,∞(μ) and from H1(μ) to L1,∞(μ). Some of the results are also new even for the classical Marcinkiewicz integral.