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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Anderson, T.C., Damián, W.: Calderón–Zygmund operators and commutators in spaces of homogeneous type: weighted inequalities. arXiv:1401.2061
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)
Bloom, S.: A commutator theorem and weighted BMO. Trans. Am. Math. Soc. 292, 103–122 (1985)
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)
Christ, M.: A \(T(b)\) theorem with remarks on analytic capacity and the Cauchy integral. Colloq. Math. 60(61), 601–628 (1990)
Chung, D.: Weighted inequalities for multivariable dyadic paraproducts. Publ. Mat. 55, 475–499 (2011)
Coifman, R., Lions, P.L., Meyer, Y., Semmes, S.: Compensated compactness and Hardy sapces. J. Math. Pures Appl. 72, 247–286 (1993)
Coifman, R.R., Rochberg, R., Weiss, G.: Factorization theorems for Hardy spaces in several variables. Ann. Math. (2) 103, 611–635 (1976)
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)
Coifman, R.R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83, 569–645 (1977)
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)
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)
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)
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)
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
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)
Folland, G.B., Stein, E.M.: Hardy Spaces on Homogeneous Groups. Princeton University Press, Princeton, NJ (1982)
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)
Guo, W., He, J., Wu, H., Yang, D.: Characterizations of the compactness of commutators associated with Lipschitz functions, arXiv:1801.06064v1
Guo, W., Lian, J., Wu, H.: The unified theory for the necessity of bounded commutators and applications. J. Geom. Anal. https://doi.org/10.1007/s12220-019-00226-y
Holmes, I., Lacey, M., Wick, B.D.: Commutators in the two-weight setting. Math. Ann. 367, 51–80 (2017)
Holmes, I., Petermichl, S., Wick, B.D.: Weighted little bmo and two-weight inequalities for Journé commutators. Anal. PDE 11, 1693–1740 (2018)
Huber, A.: On the uniqueness of generalized axially symmetric potentials. Ann. Math. 60, 351–358 (1954)
Hytönen, T.: The sharp weighted bound for general Calderón-Zygmund operators. Ann. Math. (2) 175, 1473–1506 (2012)
Hytönen, T.: The \(L^p\rightarrow L^q\) boundedness of commutators with applications to the Jacobian operator. arXiv:1804.11167
Hytönen, T., Kairema, A.: Systems of dyadic cubes in a doubling metric space. Colloq. Math. 126, 1–33 (2012)
Journé, J.L.: Calderón-Zygmund Operators, Pseudodifferential Operators and the Cauchy Integral of Calderón. Lecture Notes in Mathematics, vol. 994. Springer, Berlin (1983)
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)
Karagulyan, G.A.: An abstract theory of singular operators. Trans. Am. Math. Soc. 372, 4761–4803 (2019)
Lerner, A.K.: On pointwise estimates involving sparse operators. N. Y. J. Math. 22, 341–349 (2016)
Lerner, A.K., Nazarov, F.: Intuitive dyadic calculus: the basics. Expo. Math. 37, 225–265 (2019)
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)
Lerner, A.K., Ombrosi, S., Rivera-Ríos, I.P.: Commutators of singular integrals revisited. Bull. Lond. Math. Soc. 51, 107–119 (2019)
Li, J., Nguyen, T., Ward, L.A., Wick, B.D.: The Cauchy integral, bounded and compact commutators. Studia Math. 250, 193–216 (2020)
Macías, R.A., Segovia, C.: Lipschitz functions on spaces of homogeneous type. Adv. Math. 33, 257–270 (1979)
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)
Moen, K.: Sharp weighted bounds without testing or extrapolation. Arch. Math. (Basel) 99, 457–466 (2012)
Muckenhoupt, B., Wheeden, R.L.: Weighted bounded mean oscillation and the Hilbert transform. Studia Math. 54, 221–237 (1975/1976)
Muckenhoupt, B., Stein, E.M.: Classical expansions and their relation to conjugate harmonic functions. Trans. Am. Math. Soc. 118, 17–92 (1965)
Nehari, Z.: On bounded bilinear forms. Ann. Math. 65, 153–162 (1957)
Petermichl, S., Pott, S.: An estimate for weighted Hilbert transform via square functions. Trans. Am. Math. Soc. 354, 1699–1703 (2002)
Sawyer, E., Wheeden, R.L.: Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces. Am. J. Math. 114, 813–874 (1992)
Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton, NJ (1993)
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)
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)
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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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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DOI: https://doi.org/10.1007/s12220-019-00308-x