Two Weight Commutators on Spaces of Homogeneous Type and Applications


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).

This is a preview of subscription content, access via your institution.

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.


  1. 1.

    Anderson, T.C., Damián, W.: Calderón–Zygmund operators and commutators in spaces of homogeneous type: weighted inequalities. arXiv:1401.2061

  2. 2.

    Betancor, J.J., Harboure, E., Nowak, A., Viviani, B.: Mapping properties of fundamental operators in harmonic analysis related to Bessel operators. Stud. Math. 197, 101–140 (2010)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Bloom, S.: A commutator theorem and weighted BMO. Trans. Am. Math. Soc. 292, 103–122 (1985)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Castro, A., Szarek, T.Z.: Calderón-Zygmund operators in the Bessel setting for all possible type indices. Acta Math. Sin. (Engl. Ser.), 30, 637–648 (2014)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Christ, M.: A \(T(b)\) theorem with remarks on analytic capacity and the Cauchy integral. Colloq. Math. 60(61), 601–628 (1990)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Chung, D.: Weighted inequalities for multivariable dyadic paraproducts. Publ. Mat. 55, 475–499 (2011)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Coifman, R., Lions, P.L., Meyer, Y., Semmes, S.: Compensated compactness and Hardy sapces. J. Math. Pures Appl. 72, 247–286 (1993)

    MathSciNet  Google Scholar 

  8. 8.

    Coifman, R.R., Rochberg, R., Weiss, G.: Factorization theorems for Hardy spaces in several variables. Ann. Math. (2) 103, 611–635 (1976)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Coifman, R.R., Weiss, G.: Analyse harmonique non-commutative sur certains espaces homogènes. Étude de certaines intégrales singulières, Lecture Notes in Math. vol. 242, Springer, Berlin (1971)

  10. 10.

    Coifman, R.R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83, 569–645 (1977)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Diaz, K.P.: The Szegö kernel as a singular integral kernel on a family of weakly pseudoconvex domains. Trans. Am. Math. Soc. 304, 141–170 (1987)

    Google Scholar 

  12. 12.

    Dragiĉević, O., Grafakos, L., Pereyra, M.C., Petermichl, S.: Extrapolation and sharp norm estimates for classical operators on weighted Lebesgue spaces. Publ. Mat. 49, 73–91 (2005)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Duong, X., Holmes, I., Li, J., Wick, B.D., Yang, D.: Two weight Commutators in the Dirichlet and Neumann Laplacian settings. J. Funct. Anal. 276, 1007–1060 (2019)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Duong, X.T., Li, H.-Q., Li, J., Wick, B.D.: Lower bound for Riesz transform kernels and commutator theorems on stratified nilpotent Lie groups. J. Math. Pures Appl. (9) 124, 273–299 (2019)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Duong, X.T., Li, H.-Q., Li, J., Wick, B.D., Wu, Q.Y.: Lower bound of Riesz transform kernels revisited and commutators on stratified Lie groups. arXiv:1803.01301

  16. 16.

    Duong, X.T., Li, J., Wick, B.D., Yang, D.: Factorization for Hardy spaces and characterization for BMO spaces via commutators in the Bessel setting. Indiana Univ. Math. J. 66, 1081–1106 (2017)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Folland, G.B., Stein, E.M.: Hardy Spaces on Homogeneous Groups. Princeton University Press, Princeton, NJ (1982)

    Google Scholar 

  18. 18.

    Greiner, P.C., Stein, E.M.: On the solvability of some differential operators of type \(\Box _b\). In: Proc. Internat. Conf., (Cortona, Italy, 1976–1977), Scuola Norm. Sup. Pisa, Pisa, pp. 106–165 (1978)

  19. 19.

    Guo, W., He, J., Wu, H., Yang, D.: Characterizations of the compactness of commutators associated with Lipschitz functions, arXiv:1801.06064v1

  20. 20.

    Guo, W., Lian, J., Wu, H.: The unified theory for the necessity of bounded commutators and applications. J. Geom. Anal.

  21. 21.

    Holmes, I., Lacey, M., Wick, B.D.: Commutators in the two-weight setting. Math. Ann. 367, 51–80 (2017)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Holmes, I., Petermichl, S., Wick, B.D.: Weighted little bmo and two-weight inequalities for Journé commutators. Anal. PDE 11, 1693–1740 (2018)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Huber, A.: On the uniqueness of generalized axially symmetric potentials. Ann. Math. 60, 351–358 (1954)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Hytönen, T.: The sharp weighted bound for general Calderón-Zygmund operators. Ann. Math. (2) 175, 1473–1506 (2012)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Hytönen, T.: The \(L^p\rightarrow L^q\) boundedness of commutators with applications to the Jacobian operator. arXiv:1804.11167

  26. 26.

    Hytönen, T., Kairema, A.: Systems of dyadic cubes in a doubling metric space. Colloq. Math. 126, 1–33 (2012)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Journé, J.L.: Calderón-Zygmund Operators, Pseudodifferential Operators and the Cauchy Integral of Calderón. Lecture Notes in Mathematics, vol. 994. Springer, Berlin (1983)

  28. 28.

    Kairema, A., Li, J., Pereyra, C., Ward, L.A.: Haar bases on quasi-metric measure spaces, and dyadic structure theorems for function spaces on product spaces of homogeneous type. J. Funct. Anal. 271, 1793–1843 (2016)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Karagulyan, G.A.: An abstract theory of singular operators. Trans. Am. Math. Soc. 372, 4761–4803 (2019)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Lerner, A.K.: On pointwise estimates involving sparse operators. N. Y. J. Math. 22, 341–349 (2016)

    MathSciNet  Google Scholar 

  31. 31.

    Lerner, A.K., Nazarov, F.: Intuitive dyadic calculus: the basics. Expo. Math. 37, 225–265 (2019)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Lerner, A.K., Ombrosi, S., Rivera-Ríos, I.P.: On pointwise and weighted estimates for commutators of Calderón-Zygmund operators. Adv. Math. 319, 153–181 (2017)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Lerner, A.K., Ombrosi, S., Rivera-Ríos, I.P.: Commutators of singular integrals revisited. Bull. Lond. Math. Soc. 51, 107–119 (2019)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Li, J., Nguyen, T., Ward, L.A., Wick, B.D.: The Cauchy integral, bounded and compact commutators. Studia Math. 250, 193–216 (2020)

    MathSciNet  Article  Google Scholar 

  35. 35.

    Macías, R.A., Segovia, C.: Lipschitz functions on spaces of homogeneous type. Adv. Math. 33, 257–270 (1979)

    MathSciNet  Article  Google Scholar 

  36. 36.

    Mao, S., Wu, H., Yang, D.: Boundedness and compactness characterizations of Riesz transform commutators on Morrey spaces in the Bessel setting. Anal. Appl. 17, 145–178 (2019)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Moen, K.: Sharp weighted bounds without testing or extrapolation. Arch. Math. (Basel) 99, 457–466 (2012)

    MathSciNet  Article  Google Scholar 

  38. 38.

    Muckenhoupt, B., Wheeden, R.L.: Weighted bounded mean oscillation and the Hilbert transform. Studia Math. 54, 221–237 (1975/1976)

  39. 39.

    Muckenhoupt, B., Stein, E.M.: Classical expansions and their relation to conjugate harmonic functions. Trans. Am. Math. Soc. 118, 17–92 (1965)

    MathSciNet  Article  Google Scholar 

  40. 40.

    Nehari, Z.: On bounded bilinear forms. Ann. Math. 65, 153–162 (1957)

    MathSciNet  Article  Google Scholar 

  41. 41.

    Petermichl, S., Pott, S.: An estimate for weighted Hilbert transform via square functions. Trans. Am. Math. Soc. 354, 1699–1703 (2002)

    MathSciNet  Article  Google Scholar 

  42. 42.

    Sawyer, E., Wheeden, R.L.: Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces. Am. J. Math. 114, 813–874 (1992)

    MathSciNet  Article  Google Scholar 

  43. 43.

    Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton, NJ (1993)

    Google Scholar 

  44. 44.

    Tao, J., Yang, D., Yang, D.: Boundedness and compactness characterizations of the Cauchy integral commutators on Morrey spaces. Math. Methods Appl. Sci. 42, 1631–1651 (2019)

    MathSciNet  Article  Google Scholar 

  45. 45.

    Treil, S.: A remark on two weight estimates for positive dyadic operators. In: Operator-Related Function Theory and Time-Frequency Analysis, Abel Symp., vol. 9, pp. 185–195. Springer, Cham (2015)

Download references


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).

Author information



Corresponding author

Correspondence to Dongyong Yang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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).

Download citation


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

Mathematics Subject Classification

  • 42B30
  • 42B20
  • 42B35