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The Hardy Space \({H}^{1}(\mathcal{X},\,\nu )\) and Its Dual Space \(\mathrm{RBMO}(\mathcal{X},\nu )\)

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

Abstract

In this chapter, we introduce and study a class of metric measure spaces \((\mathcal{X},d,\nu )\), which include both Euclidean spaces with nonnegative Radon measures satisfying the polynomial growth condition and spaces of homogeneous type as special cases. We also introduce the BMO-type space \(\mathrm{RBMO}\,(\mathcal{X},\,\nu )\) and the atomic Hardy space \({H}^{1}(\mathcal{X},\,\nu )\) in this setting, establish the John–Nirenberg inequality for \(\mathrm{RBMO}\,(\mathcal{X},\,\nu )\) and some equivalent characterizations of \(\mathrm{RBMO}\,(\mathcal{X},\,\nu )\) and \({H}^{1}(\mathcal{X},\,\nu )\), respectively, and show that the dual space of \({H}^{1}(\mathcal{X},\,\nu )\) is \(\mathrm{RBMO}\,(\mathcal{X},\,\nu )\).

Keywords

  • Measure Space
  • Homogeneous Type
  • Doubling Condition
  • Doubling Measure
  • Equivalent Characterization

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 [18, p. 67].

  2. 2.

    See [92] and [148].

  3. 3.

    See [49, Theorem 1.2].

  4. 4.

    See [19].

  5. 5.

    See [19].

  6. 6.

    See [110, Theorem 4.3].

  7. 7.

    See [110, Theorem 4.13].

  8. 8.

    See [110, Corollary 2.12 (b)].

  9. 9.

    See [19, p. 593, Theorem B].

  10. 10.

    See [19].

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Yang, D., Yang, D., Hu, G. (2013). The Hardy Space \({H}^{1}(\mathcal{X},\,\nu )\) and Its Dual Space \(\mathrm{RBMO}(\mathcal{X},\nu )\) . 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_7

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