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Boundedness of Operators over \((\mathcal{X},\nu )\)

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

Abstract

In this chapter, we consider the boundedness of Calderón–Zygmund operators over non-homogeneous spaces \((\mathcal{X},\nu )\). We first show that the Calderón–Zygmund operator T is bounded from \({H}^{1}(\mathcal{X},\nu )\) to \({L}^{1}(\mathcal{X},\,\nu )\). We then establish the molecular characterization of a version of the atomic Hardy space \(\tilde{H}_{\mathrm{atb}}^{1,\,p}(\mathcal{X},\nu )\), which is a subspace of \(H_{\mathrm{atb}}^{1,\,p}(\mathcal{X},\nu )\), and obtain the boundedness of T on \(\tilde{H}_{\mathrm{atb}}^{1,\,p}(\mathcal{X},\nu )\). We also prove that the boundedness of T on \({L}^{p}(\mathcal{X},\,\nu )\) with p ∈ (1, ) is equivalent to its various estimates and establish some weighted estimates involving the John–Strömberg maximal operators and the John–Strömberg sharp maximal operators, and some weighted norm inequalities for the multilinear Calderón–Zygmund operators. In addition, the boundedness of multilinear commutators of Calderón–Zygmund operators on Orlicz spaces is also presented.

Keywords

  • Sharp Maximal Operator
  • Atomic Hardy Space
  • Multilinear Commutators
  • Weighted Norm Inequalities
  • Marcinkiewicz Integral

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 [127] or [70].

  2. 2.

    See [40, p. 65].

  3. 3.

    See [70, Lemma 10.1].

  4. 4.

    See [111, Theorem 6.4].

  5. 5.

    See, for example, [19].

  6. 6.

    See [126].

  7. 7.

    See, for example, [49, p. 88] and [156, (2.1)].

  8. 8.

    See [93] for more properties of \(a_{\Phi }\) and \(b_{\Phi }\).

  9. 9.

    See [83, Theorem 1.13].

  10. 10.

    See [4, Lemma 3.4].

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Yang, D., Yang, D., Hu, G. (2013). Boundedness of Operators over \((\mathcal{X},\nu )\) . 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_8

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