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Science China Mathematics

, Volume 57, Issue 1, pp 123–144 | Cite as

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

  • HaiBo Lin
  • DaChun Yang
Articles

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 fin 1,∞ (µ) into L 1(µ). These results essentially improve the known results even for non-doubling measures.

Keywords

upper doubling geometrically doubling Marcinkiewicz integral atomic Hardy space RBMO(µ) 

MSC(2010)

42B20 42B25 42B35 30L99 

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Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.College of ScienceChina Agricultural UniversityBeijingChina
  2. 2.School of Mathematical SciencesBeijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of EducationBeijingChina

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