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

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  • First Online:
The Hardy Space H1 with Non-doubling Measures and Their Applications

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

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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.

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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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Authors and Affiliations

  1. School of Mathematical Sciences, Beijing Normal University Laboratory of Mathematics and Complex Systems Ministry of Education, Beijing, People’s Republic of China

    Dachun Yang

  2. School of Mathematical Sciences, Xiamen University, Xiamen, People’s Republic of China

    Dongyong Yang

  3. Department of Applied Mathematics, Zhengzhou Information Science and Technology Institute, Zhengzhou, People’s Republic of China

    Guoen Hu

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  1. Dachun Yang
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  2. Dongyong Yang
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  3. Guoen Hu
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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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