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Boundedness of certain commutators over non-homogeneous metric measure spaces

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Abstract

Let \((\mathcal {X},d,\mu )\) be a metric measure space satisfying the so-called upper doubling condition and the geometrically doubling condition. Let T be a Calderón-Zygmund operator with kernel satisfying only the size condition and some Hörmander-type condition, and \(b\in \widetilde{\mathrm{RBMO}}(\mu )\) (the regularized BMO space with the discrete coefficient). In this paper, the authors establish the boundedness of the commutator \(T_b:=bT-Tb\) generated by T and b from the atomic Hardy space \(\widetilde{H}^1(\mu )\) with the discrete coefficient into the weak Lebesgue space \(L^{1,\,\infty }(\mu )\). From this and an interpolation theorem for sublinear operators which is also proved in this paper, the authors further show that the commutator \(T_b\) is bounded on \(L^p(\mu )\) for all \(p\in (1,\infty )\). Moreover, the boundedness of the commutator generated by the generalized fractional integral \(T_\alpha \,(\alpha \in (0,1))\) and the \(\widetilde{\mathrm{RBMO}}(\mu )\) function from \(\widetilde{H}^1(\mu )\) into \(L^{1/{(1-\alpha )},\,\infty }(\mu )\) is also presented.

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Authors and Affiliations

  1. College of Science, China Agricultural University, Beijing, 100083, People’s Republic of China

    Haibo Lin

  2. School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, 100875, People’s Republic of China

    Suqing Wu & Dachun Yang

Authors
  1. Haibo Lin
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  2. Suqing Wu
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  3. Dachun Yang
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Correspondence to Dachun Yang.

Additional information

Haibo Lin is supported by the National Natural Science Foundation of China (Grant Nos. 11301534 and 11471042) and Da Bei Nong Education Fund (Grant No. 1101-2413002). Dachun Yang is supported by the National Natural Science Foundation of China (Grant Nos. 11571039 and 11361020) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20120003110003).

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Lin, H., Wu, S. & Yang, D. Boundedness of certain commutators over non-homogeneous metric measure spaces. Anal.Math.Phys. 7, 187–218 (2017). https://doi.org/10.1007/s13324-016-0136-6

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