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The Local Atomic Hardy Space h 1(μ)

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Part of the Lecture Notes in Mathematics book series (LNM,volume 2084)

Abstract

This chapter is mainly devoted to the study of the local version of H 1(μ) and its dual space. First, we introduce a local atomic Hardy space h 1(μ) and a local BMO-type space rbmo (μ). After presenting some basic properties of these spaces, we then prove that the space rbmo (μ) satisfies the John–Nirenberg inequality and its predual space is h 1(μ). Moreover, we also establish the relations between H 1(μ) and h 1(μ) as well as between RBMO (μ) and rbmo (μ). In addition, we also introduce a BLO-type space RBLO (μ) and its local version rblo (μ) on \(({\mathbb{R}}^{D},\vert \cdot \vert,\mu )\) with μ as in (0.0.1) and establish some characterizations of both RBLO (μ) and rblo (μ).

Keywords

  • Local Hardy Space
  • Space RBMO
  • John-Nirenberg Inequality
  • Predual Space
  • Initial Cube

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

  1. 1.

    See [38].

  2. 2.

    See [38].

  3. 3.

    See [38].

  4. 4.

    See [39, pp. 294–296].

  5. 5.

    See [17].

References

  1. C. Bennett, Another characterization of BLO. Proc. Am. Math. Soc. 85, 552–556 (1982)

    Google Scholar 

  2. C. Bennett, R.A. DeVore, R. Sharpley, Weak-L and BMO. Ann. of Math. (2) 113, 601–611 (1981)

    Google Scholar 

  3. R.R. Coifman, R. Rochberg, Another characterization of BMO. Proc. Am. Math. Soc. 79, 249–254 (1980)

    CrossRef  MATH  MathSciNet  Google Scholar 

  4. D. Goldberg, A local version of real Hardy spaces. Duke Math. J. 46, 27–42 (1979)

    CrossRef  MATH  MathSciNet  Google Scholar 

  5. L. Grafakos, Estimates for maximal singular integrals. Colloq. Math. 96, 167–177 (2003)

    CrossRef  MATH  MathSciNet  Google Scholar 

  6. G. Hu, Da. Yang, Do. Yang, h 1, bmo, blo and Littlewood–Paley g-functions with non-doubling measures. Rev. Mat. Iberoam. 25, 595–667 (2009)

    Google Scholar 

  7. Y. Jiang, Spaces of type BLO for non-doubling measures. Proc. Am. Math. Soc. 133, 2101–2107 (2005)

    CrossRef  MATH  Google Scholar 

  8. W. Ou, The natural maximal operator on BMO. Proc. Am. Math. Soc. 129, 2919–2921 (2001)

    CrossRef  MATH  Google Scholar 

  9. D. Yang, Local Hardy and BMO spaces on non-homogeneous spaces. J. Aust. Math. Soc. 79, 149–182 (2005)

    CrossRef  MATH  MathSciNet  Google Scholar 

  10. Da. Yang, Do. Yang, Uniform boundedness for approximations of the identity with non-doubling measures. J. Inequal. Appl. Art. ID 19574, 25 pp. (2007)

    Google Scholar 

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Yang, D., Yang, D., Hu, G. (2013). The Local Atomic Hardy Space h 1(μ). In: The Hardy Space H1 with Non-doubling Measures and Their Applications. Lecture Notes in Mathematics, vol 2084. Springer, Cham. https://doi.org/10.1007/978-3-319-00825-7_4

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