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Commutators of maximal functions on spaces of homogeneous type and their weighted, local versions

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Abstract

We obtain the characterizations of commutators of several versions of maximal functions on spaces of homogeneous type. In addition, with the aid of interpolation theory, we provide weighted version of the commutator theorems by establishing new characterizations of the weighted BMO space. Finally, a concrete example shows that the local version of commutators also has an independent interest.

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References

  1. Agcayazi M, Gogatishvili A, Koca K, Mustafayev R. A note on maximal commutators and commutators of maximal functions. J Math Soc Japan, 2015, 67(2): 581–593

    Article  MathSciNet  MATH  Google Scholar 

  2. Aimar H, Bernardis A, Iaffei B. Multiresolution approximations and unconditional bases on weighted Lebesgue spaces on spaces of homogeneous type. J Approx Theory, 2007, 148(1): 12–34

    Article  MathSciNet  MATH  Google Scholar 

  3. Auscher P, Hytönen T. Orthonormal bases of regular wavelets in spaces of homogeneous type. Appl Comput Harmon Anal, 2013, 34(2): 266–296

    Article  MathSciNet  MATH  Google Scholar 

  4. Bastero J, Milman M, Ruiz F J. Commutators for the maximal and sharp functions. Proc Amer Math Soc, 2000, 128: 3329–3334

    Article  MathSciNet  MATH  Google Scholar 

  5. Carbery A, Chang S-Y A, Garnett J. Weights and L log L. Pacific J Math, 1985, 120(1): 33–45

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen P, Li J, Ward L A. BMO from dyadic BMO via expectations on product spaces of homogeneous type. J Funct Anal, 2013, 265: 2420–2451

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen W, Sawyer E. Endpoint estimates for commutators of singular integrals on spaces of homogeneous type. J Math Anal Appl, 2003, 282(2): 553–566

    Article  MathSciNet  MATH  Google Scholar 

  8. 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  MATH  Google Scholar 

  9. Coifman R R, Weiss G. Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes. Lecture Notes in Math, Vol 242. Berlin: Springer-Verlag, 1971

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. David G, Journé J L, Semmes S. Calderón-Zygmund operators, para-accretive functions and interpolation. Rev Mat Iberoam, 1985, 1(4): 1–56

    Article  MATH  Google Scholar 

  12. Deng D G, Han Y S. Harmonic Analysis on Spaces of Homogeneous Type. Berlin: Springer, 2009

    Book  MATH  Google Scholar 

  13. Duong X T, Lacey M, Li J, Wick B D, Wu Q Y. Commutators of Cauchy type integrals for domains in ℂn with minimal smoothness. Indiana Univ Math J (in press)

  14. Edmunds D E, Kokilashvili V, Meskhi A. Weight inequalities for singular integrals defined on spaces of homogeneous and nonhomogeneous type. Georgian Math J, 2001, 8(1): 33–59

    Article  MathSciNet  MATH  Google Scholar 

  15. García-Cuerva J, Harboure E, Segovia C, Torrea J L. Weighted norm inequalities for commutators of strongly singular integrals. Indiana Univ Math J, 1991, 40: 1397–1420

    Article  MathSciNet  MATH  Google Scholar 

  16. García-Cuerva J, Rubio de Francia J L. Weighted Norm Inequalities and Related Topics. North-Holland Math Studies, Vol 116. Amsterdam: North Holland, 1985

    Book  MATH  Google Scholar 

  17. Grafakos L, Liu L G, Yang D C. Vector-valued singular integrals and maximal functions on spaces of homogeneous type. Math Scand, 2009, 104(2): 296–310

    Article  MathSciNet  MATH  Google Scholar 

  18. Han Y S. Calderón-type reproducing formula and the Tb theorem. Rev Mat Iberoam, 1994, 10: 51–91

    Article  MATH  Google Scholar 

  19. Han Y S. Plancherel-Pôlya type inequality on spaces of homogeneous type and its applications. Proc Amer Math Soc, 1998, 126(11): 3315–3327

    Article  MathSciNet  MATH  Google Scholar 

  20. Han Y S, Li J, Lin C C. Criterions of the L2 boundedness and sharp endpoint estimates for singular integral operators on product spaces of homogeneous type. Ann Sc Norm Super Pisa Cl Sci, 2016, 16(3): 845–907

    MathSciNet  MATH  Google Scholar 

  21. Han Y S, Sawyer E T. Littlewood-Paley Theory on Spaces of Homogeneous Type and the Classical Function Spaces. Mem Amer Math Soc, Vol 110, No 530. Providence: Amer Math Soc, 1994

    MATH  Google Scholar 

  22. Hart J, Torres R H. John-Nirenberg inequalities and weight invariant BMO spaces. J Geom Anal, 2019, 29(2): 1608–1648

    Article  MathSciNet  MATH  Google Scholar 

  23. He Z Y, Han Y S, Li J, Liu L G, Yang D C, Yuan W. A complete real-variable theory of Hardy spaces on spaces of homogeneous type. J Fourier Anal Appl, 2019, 25(5): 2197–2267

    Article  MathSciNet  MATH  Google Scholar 

  24. Hu G E, Lin H B, Yang D C. Commutators of the Hardy-Littlewood maximal operator with BMO symbols on spaces of homogeneous type. Abstr Appl Anal, 2008, 2008(Mar): (21 pp)

  25. Hu G E, Yang D C. Maximal commutators of BMO functions and singular integral operators with non-smooth kernels on spaces of homogeneous type. J Math Anal Appl, 2009, 354(1): 249–262

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  27. Krantz S G, Li S Y. Boundedness and compactness of integral operators on spaces of homogeneous type and applications, II. J Math Anal Appl, 2001, 258: 642–657

    Article  MathSciNet  MATH  Google Scholar 

  28. Kronz M. Some function spaces on spaces of homogeneous type. Manuscripta Math, 2001, 106(2): 219–248

    Article  MathSciNet  MATH  Google Scholar 

  29. Lanzani L, Stein E M. The Cauchy-Szegö projection for domains in ℂn with minimal smoothness. Duke Math J, 2017, 166: 125–176

    Article  MathSciNet  MATH  Google Scholar 

  30. Lerner A K. On weighted estimates of non-increasing rearrangements. East J Approx, 1998, 4: 277–290

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  32. Milman M, Schonbek T. Second order estimates in interpolation theory and applications. Proc Amer Math Soc, 1990, 110(4): 961–969

    Article  MathSciNet  MATH  Google Scholar 

  33. Nagel A, Stein E M. On the product theory of singular integrals. Rev Mat Iberoam, 2004, 20: 531–561

    Article  MathSciNet  MATH  Google Scholar 

  34. Nagel A, Stein E M. The \(\bar{\partial}_{b}\)-complex on decoupled boundaries in ℂn. Ann of Math (2), 2006, 164(2): 649–713

    Article  MathSciNet  MATH  Google Scholar 

  35. Pradolini G, Salinas O. Commutators of singular integrals on spaces of homogeneous type. Czechoslovak Math J, 2007, 57(1): 75–93

    Article  MathSciNet  MATH  Google Scholar 

  36. Range R M. Holomorphic Functions and Integral Representations in Several Complex Variables. Grad Texts in Math, Vol 108. New York: Springer-Verlag, 1986

    Google Scholar 

  37. Rao M M, Ren Z D. Theory of Orlicz Spaces. New York: Marcel Dekker, 1991

    MATH  Google Scholar 

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Acknowledgements

This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 12171221, 12071197) and the Natural Science Foundation of Shandong Province (Grant Nos. ZR2019YQ04, 2020KJI002).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Linyi University, Linyi, 276005, China

    Zunwei Fu & Qingyan Wu

  2. School of Mathematical Sciences, Qufu Normal University, Qufu, 273100, China

    Zunwei Fu

  3. Department of Mathematics and Statistics, Saint Louis University, St. Louis, MO, 63103, USA

    Elodie Pozzi

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  1. Zunwei Fu
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  2. Elodie Pozzi
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  3. Qingyan Wu
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Correspondence to Qingyan Wu.

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Fu, Z., Pozzi, E. & Wu, Q. Commutators of maximal functions on spaces of homogeneous type and their weighted, local versions. Front. Math. China 17, 625–652 (2022). https://doi.org/10.1007/s11464-021-0912-y

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