Two Weight Commutators on Spaces of Homogeneous Type and Applications

Abstract

In this paper, we establish the two weight commutator theorem of Calderón–Zygmund operators in the sense of Coifman–Weiss on spaces of homogeneous type, by studying the weighted Hardy and BMO space for \(A_2\) weights and by proving the sparse operator domination of commutators. The main tool here is the Haar basis, the adjacent dyadic systems on spaces of homogeneous type, and the construction of a suitable version of a sparse operator on spaces of homogeneous type. As applications, we provide a two weight commutator theorem (including the high order commutators) for the following Calderón–Zygmund operators: Cauchy integral operator on \({\mathbb {R}}\), Cauchy–Szegö projection operator on Heisenberg groups, Szegö projection operators on a family of unbounded weakly pseudoconvex domains, the Riesz transform associated with the sub-Laplacian on stratified Lie groups, as well as the Bessel Riesz transforms (in one and several dimensions).

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Acknowledgements

The authors would like to thank the referees for careful reading and helpful suggestions, which helped to make this paper more accurate and readable. X. T. Duong and J. Li are supported by the Australian Research Council (ARC) through the research grants DP 190100970 and DP 170101060, respectively, and also supported by Macquarie University Research Seeding Grant. B. D. Wick’s research supported in part by National Science Foundation DMS grant #1560995 and # 1800057. R. M. Gong is supported by NNSF of China (Grant No. 11401120) and the Foundation for Distinguished Young Teachers in Higher Education of Guangdong Province (Grant No. YQ2015126). D. Yang is supported by the NNSF of China (Grant Nos. 11971402 and 11871254).

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Correspondence to Dongyong Yang.

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Duong, X.T., Gong, R., Kuffner, MJ.S. et al. Two Weight Commutators on Spaces of Homogeneous Type and Applications. J Geom Anal 31, 980–1038 (2021). https://doi.org/10.1007/s12220-019-00308-x

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Keywords

  • BMO
  • Commutator
  • Two weights
  • Hardy space
  • Factorization

Mathematics Subject Classification

  • 42B30
  • 42B20
  • 42B35