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Commutators of Littlewood-Paley operators

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Abstract

Let b ∈ L loc(ℝn) and L denote the Littlewood-Paley operators including the Littlewood-Paley g function, Lusin area integral and g *λ function. In this paper, the authors prove that the L p boundedness of commutators [b, L] implies that b ∈ BMO(ℝn). The authors therefore get a characterization of the L p-boundedness of the commutators [b, L]. Notice that the condition of kernel function of L is weaker than the Lipshitz condition and the Littlewood-Paley operators L is only sublinear, so the results obtained in the present paper are essential improvement and extension of Uchiyama’s famous result.

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Authors and Affiliations

  1. Department of Mathematics and Mechanics, Applied Science School, University of Science and Technology Beijing, Beijing, 100083, China

    YanPing Chen

  2. School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

    Yong Ding

  3. Laboratory of Mathematics and Complex Systems (BNU), Ministry of Education, Beijing, 100083, China

    Yong Ding

Authors
  1. YanPing Chen
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  2. Yong Ding
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Correspondence to Yong Ding.

Additional information

This work was partially supported by National Natural Science Foundation of China (Grant No. 10931001, 10826046) and Specialized Research Foundation for Doctor Programme (Grant No. 20050027025)

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Chen, Y., Ding, Y. Commutators of Littlewood-Paley operators. Sci. China Ser. A-Math. 52, 2493–2505 (2009). https://doi.org/10.1007/s11425-009-0178-4

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  • DOI: https://doi.org/10.1007/s11425-009-0178-4

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