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Equivalent boundedness of Marcinkiewicz integrals on non-homogeneous metric measure spaces

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Abstract

Let (X, d,µ) be a metric measure space satisfying the upper doubling condition and the geometrically doubling condition in the sense of Hytönen. We prove that the L p(µ)-boundedness with p ∈ (1,∞) of the Marcinkiewicz integral is equivalent to either of its boundedness from L 1(µ) into L 1,∞(µ) or from the atomic Hardy space H 1(µ) into L 1(µ). Moreover, we show that, if the Marcinkiewicz integral is bounded from H 1(µ) into L 1(µ), then it is also bounded from L ∞(µ) into the space RBLO(µ) (the regularized BLO), which is a proper subset of RBMO(µ) (the regularized BMO) and, conversely, if the Marcinkiewicz integral is bounded from L ∞ b (µ) (the set of all L ∞(µ) functions with bounded support) into the space RBMO(µ), then it is also bounded from the finite atomic Hardy space H 1,∞fin (µ) into L 1(µ). These results essentially improve the known results even for non-doubling measures.

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Authors and Affiliations

  1. College of Science, China Agricultural University, Beijing, 100083, China

    HaiBo Lin

  2. School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, 100875, China

    DaChun Yang

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  1. HaiBo Lin
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  2. DaChun Yang
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Correspondence to DaChun Yang.

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Lin, H., Yang, D. Equivalent boundedness of Marcinkiewicz integrals on non-homogeneous metric measure spaces. Sci. China Math. 57, 123–144 (2014). https://doi.org/10.1007/s11425-013-4754-2

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