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Weighted estimates for iterated commutators of multilinear Calderón-Zygmund operators on non-homogeneous metric measure spaces

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Abstract

Let (\({\cal X}\), d,μ) be a non-homogeneous metric measure space satisfying the so-called upper doubling and geometrically doubling conditions, which includes the space of homogeneous type and the Euclidean space with the non-doubling measure as special cases. Let T be a multilinear Calderón-Zygmund operator and \(\vec b: = ({b_1}, \ldots ,{b_m})\) be a finite family of \(\widetilde{{\rm{RBMO}}}(\mu)\) functions. In this paper, some weak-type multiple weighted estimates for the iterated commutator \({{\rm{T}}_{\Pi \vec b}}\) generated by T and \(\vec b\) are obtained.

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References

  1. Bui T A, Duong X T. Hardy spaces, regularized BMO spaces and the boundedness of Calderón-Zygmund operators on non-homogeneous spaces. J Geom Anal, 2013, 23: 895–932

    Article  MathSciNet  Google Scholar 

  2. Calderón A P. Commutators of singular integral operators. Proc Natl Acad Sci USA, 1965, 53: 1092–1099

    Article  MathSciNet  Google Scholar 

  3. Chen J, Lin H. Hardy-type space estimates for multilinear commutators of Calderón-Zygmund operators on non-homogeneous metric measure spaces. Banach J Math Anal, 2017, 11: 477–496

    Article  MathSciNet  Google Scholar 

  4. Coifman R R, Meyer Y. On commutators of singular integrals and bilinear singular integrals. Trans Amer Math Soc, 1975, 212: 315–331

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. Coifman R R, Weiss G. Analyse harmonique non-commutative sur certains espaces homogenes. Lecture Notes in Mathematics, vol. 242. Berlin-New York: Springer-Verlag, 1971

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Fu X, Lin H, Yang D C, et al. Hardy spaces Hp over non-homogeneous metric measure spaces and their applications. Sci China Math, 2015, 58: 309–388

    Article  MathSciNet  Google Scholar 

  9. Fu X, Yang D C, Yang D Y. The molecular characterization of the Hardy space H1 on non-homogeneous metric measure spaces and its application. J Math Anal Appl, 2010, 410: 1028–1042

    Article  Google Scholar 

  10. Fu X, Yang D, Yuan W. Boundedness of multilinear commutators of Calderón-Zygmund operators on Orlicz spaces over non-homogeneous spaces. Taiwanese J Math, 2012, 16: 2203–2238

    Article  MathSciNet  Google Scholar 

  11. Grafakos L. Modern Fourier Analysis, 2nd ed. Graduate Texts in Mathematics, vol. 250. New York: Springer, 2009

    Book  Google Scholar 

  12. Grafakos L, Liu L, Maldonado D, et al. Multilinear analysis on metric spaces. Dissertationes Math (Rozprawy Mat), 2014, 497: 1–121

    Article  MathSciNet  Google Scholar 

  13. Grafakos L, Torres R H. Multilinear Calderén-Zygmund theory. Adv Math, 2002, 165: 124–164

    Article  MathSciNet  Google Scholar 

  14. Han Y, Müller D, Yang D. A Theory of Besov and Triebel-Lizorkin spaces on metric measure spaces modeled on Carnot-Carathéodory spaces. Abstr Appl Anal, 2008, 2008: 893409

    Article  Google Scholar 

  15. Heinenon J. Lectures on Analysis on Metric Spaces. New York: Springer-Verlag, 2001

    Book  Google Scholar 

  16. Hu G, Li K. Weighted vector-valued inequalities for a class of multilinear singular integral operators. Proc Edinb Math Soc (2), 2018, 61: 413–436

    Article  MathSciNet  Google Scholar 

  17. Hu G, Meng Y, Yang D. Weighted norm inequalities for multilinear Calderón-Zygmund operators on non-homogeneous metric measure spaces. Forum Math, 2014, 26: 1289–1322

    Article  MathSciNet  Google Scholar 

  18. Hu G, Yang D C, Yang D Y. Weighted estimates for commutators of singular integral operators with non-doubling measures. Sci China Ser A, 2007, 50: 1621–1641

    Article  MathSciNet  Google Scholar 

  19. Hytönen T. A framework for non-homogeneous analysis on metric spaces, and the RBMO space of Tolsa. Publ Mat, 2010, 54: 485–504

    Article  MathSciNet  Google Scholar 

  20. Hytönen T, Martikainen H. Non-homogeneous Tb theorem and random dyadic cubes on metric measure spaces. J Geom Anal, 2012, 22: 1071–1107

    Article  MathSciNet  Google Scholar 

  21. Hytönen T, Yang D C, Yang D Y. The Hardy space H1 on non-homogeneous metric spaces. Math Proc Cambridge Philos Soc, 2012, 153: 9–31

    Article  MathSciNet  Google Scholar 

  22. Komori Y. Weighted estimates for operators generated by maximal functions on nonhomogeneous spaces. Georgian Math J, 2005, 12: 121–130

    MathSciNet  MATH  Google Scholar 

  23. Lerner A, Ombrosi S, Pérez C, et al. New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory. Adv Math, 2009, 220: 1222–1264

    Article  MathSciNet  Google Scholar 

  24. Li K, Sun W. Weak and strong type weighted estimates for multilinear Calderón-Zygmund operators. Adv Math, 2014, 254: 736–771

    Article  MathSciNet  Google Scholar 

  25. Lin H, Meng Y, Yang D. Weighted estimates for commutators of multilinear Calderón-Zygmund operators with non-doubling measures. Acta Math Sci Ser B Engl Ed, 2010, 30: 1–18

    MathSciNet  MATH  Google Scholar 

  26. Lin H, Wu S, Yang D. Boundedness of certain commutators over non-homogeneous metric measure spaces. Anal Math Phys, 2017, 7: 187–218

    Article  MathSciNet  Google Scholar 

  27. Lin H, Yang D. Equivalent boundedness of Marcinkiewicz integrals on non-homogeneous metric measure spaces. Sci China Math, 2014, 57: 123–144

    Article  MathSciNet  Google Scholar 

  28. Muckenhoupt B. Weighted norm inequalities for the Hardy maximal function. Trans Amer Math Soc, 1972, 165: 207–226

    Article  MathSciNet  Google Scholar 

  29. Orobitg J, Pérez C. Ap weights for nondoubling measures in ℝn and applications. Trans Amer Math Soc, 2002, 345: 2013–2033

    Article  Google Scholar 

  30. Pérez C, Pradolini G, Torres R H, et al. End-point estimates for iterated commutators of multilinear singular integrals. Bull Lond Math Soc, 2014, 46: 26–42

    Article  MathSciNet  Google Scholar 

  31. Rao M, Ren Z. Theory of Orlicz Spaces. New York: Dekker, 1991

    MATH  Google Scholar 

  32. Sawano Y, Shimomura T, Tanaka H. A remark on modified Morrey spaces on metric measure spaces. Hokkaido Math J, 2018, 47: 1–15

    Article  MathSciNet  Google Scholar 

  33. Strömberg J O, Torchinsky A. Weighted Hardy Spaces. Lecture Notes in Mathematics, vol. 1381. Berlin: Springer-Verlag, 1989

    Book  Google Scholar 

  34. Tan C Q, Li J. Littlewood-Paley theory on metric measure spaces with non doubling measures and its applications. Sci China Math, 2015, 58: 983–1004

    Article  MathSciNet  Google Scholar 

  35. Tolsa X. BMO, H1, and Calderón-Zygmund operators for non doubling measures. Math Ann, 2001, 319: 89–149

    Article  MathSciNet  Google Scholar 

  36. Tolsa X. Painlevé’s problem and the semiadditivity of analytic capacity. Acta Math, 2003, 190: 105–149

    Article  MathSciNet  Google Scholar 

  37. Tolsa X. Analytic capacity, the Cauchy transform, and non-homogeneous Calderón-Zygmund theory. Progr Math, 2014, 307: 239–271

    MATH  Google Scholar 

  38. Volberg A, Wick B D. Bergman-type singular operators and the characterization of Carleson measures for Besov-Sobolev spaces on the complex ball. Amer J Math, 2012, 134: 949–992

    Article  MathSciNet  Google Scholar 

  39. Yan Y, Chen J, Lin H. Weighted Morrey spaces on non-homogeneous metric measure spaces. J Math Anal Appl, 2017, 452: 335–350

    Article  MathSciNet  Google Scholar 

  40. Yang D C, Yang D Y, Hu G. The Hardy Space H1 with Non-Doubling Measures and Their Applications. Lecture Notes in Mathematics, vol. 2084. Berlin: Springer-Verlag, 2013

    Book  Google Scholar 

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11301534 and 11571039). The authors express their deep gratitude to the referees for their carefully reading and helpful remarks, which indeed improved the presentation of this paper.

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Authors and Affiliations

  1. College of Science, China Agricultural University, Beijing, 100083, China

    Yuan Zhao & Haibo Lin

  2. School of Mathematics, Renmin University of China, Beijing, 100872, China

    Yan Meng

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  1. Yuan Zhao
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  2. Haibo Lin
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  3. Yan Meng
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Correspondence to Haibo Lin.

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Zhao, Y., Lin, H. & Meng, Y. Weighted estimates for iterated commutators of multilinear Calderón-Zygmund operators on non-homogeneous metric measure spaces. Sci. China Math. 64, 519–546 (2021). https://doi.org/10.1007/s11425-018-9471-7

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