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The Unified Theory for the Necessity of Bounded Commutators and Applications

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Abstract

This paper gives unified criterions on the necessity of bounded commutators in linear and multilinear settings. Our results relax the restriction of Banach spaces in previous results to quasi-Banach spaces and extend \(BMO(\mathbb {R}^n)\) to the general \(BMO_\mu \), which includes \(BMO(\mathbb {R}^n)\), \(\mathrm{Lip}_\beta (\mathbb {R}^n)\), and their weighted versions. Moreover, the conditions of kernels are also essentially weakened. As applications, some necessary conditions for bounded commutators, which are new in the endpoint case, and several new characterizations of BMO spaces, Lipschitz spaces, and their weighted versions via boundedness of commutators in various function spaces are deduced.

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References

  1. Accomazzo, N.: A characterization of BMO in terms of endpoint bounds for commutators of singular integrals. Isr. J. Math. 228(2), 787–800 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  2. Anh, B.T., Duong, X.T.: On commutators of vecter \(BMO\) functions and multilinear singular integrals with non-smooth kernels. J. Math. Anal. Appl. 371, 80–94 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bloom, S.: A commutator theorem and weighted BMO. Trans. Am. Math. Soc. 292(1), 103–122 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  4. Caldéron, A.P.: Commutators of singular integral operators. Proc. Natl. Acad. Sci. USA 53, 1092–1099 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  5. Caldéron, A.P., Weiss, M., Zygmund, A.: On the existence of singular integrals. In: Proceedings of Symposia in Pure Mathematics, vol. 10, pp. 56–73. American Mathematical Society, Providence, RI (1967)

  6. Chaffee, L.: Characterizations of bounded mean oscillation through commutators of bilinear singular integral operators. Proc. R. Soc. Edinb. 146A, 1159–1166 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chaffee, L., Cruz-Uribe, D.: Necessary conditions for the boundedness of linear and bilinear commutators on Banach function spaces. Math. Inequal. Appl. 21(1), 1–16 (2018)

    MathSciNet  MATH  Google Scholar 

  8. Chanillo, S.: A note on commutators. Indiana Univ. Math. J. 31, 7–16 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, S., Wu, H.: Multiple weighted estimates for commutators of multilinear singular integrals with non-smooth kernels. J. Math. Anal. Appl. 396, 888–903 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen, S., Wu, H.: Multiple weighted estimates for commutators of multilinear fractional integral operators. Sci. China Math. 56(9), 1879–1894 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, Y., Ding, Y., Hong, G.: Commutators with fractional differention and new characterizations of BMO-Sobolev spaces. Anal. PDE 9(6), 1497–1522 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen, X., Xue, Q.: Weighted estimates for a class of multilinear fractional type operators. J. Math. Anal. Appl. 362, 355–373 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Cruz-Uribe, D., Moen, K.: A multilinear reverse Hölder inequality with applications to multilinear weighted norm inequalities. arXiv:1701.07800 (2017)

  15. Ding, Y.: Higher order commutators for a class of rough operators. Ark. Mat. 37, 33–44 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  16. Grafakos, L., Torres, R.H.: Multilinear Calderon–Zygmund theory. Adv. Math. 165, 124–164 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  17. Holmes, I., Lacey, M.T., Spencer, S.: Commutatots with fractional integral operators. Studia Math. 233(3), 279–291 (2016)

    MathSciNet  Google Scholar 

  18. Holmes, I., Lacey, M.T., Wick, B.D.: Commutators in the two weight setting. Math. Ann. 367(1–2), 51–80 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hu, G.: \(L^p(\mathbb{R}^n)\) boundedness of the commutator of a homogeneous singular integral operator. Studia Math. 154(1), 13–27 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hu, B., Gu, J.: Necessary and sufficient conditions for boundedness of some commutators with weighted Lipschitz functions. J. Math. Anal. Appl. 340(1), 598–605 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Janson, S.: Mean oscillation and commutators of singular integral operators. Ark. Mat. 16, 263–270 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lerner, A., Ombrosi, S., Pérez, C., Torres, R., Trujillo-González, R.: New maximal functions and multiple weights for the multilinear Calderón–Zygmund theory. Adv. Math. 220(4), 1222–1264 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lerner, A.K., Ombrasi, S., Rivera-Ríos, I.P.: Commutators of singular integrals revisited. arXiv: 1709.04724v1 (2017)

  24. Li, J., Wick, B.D.: Weak factorizations of the Hardy space \(H^1(\mathbb{R}^n)\) in terms of multilinear Riesz transform. Can. Math. Bull. 60(3), 571–585 (2017)

    Article  MATH  MathSciNet  Google Scholar 

  25. Lin, Y., Liu, Z., Pan, G.: Weighted Lipschitz estimates for commutators of fractional integrals with homogeneous kernels. Taiwan. J. Math. 15(6), 2689–2700 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Liu, L., Zhou, X.: Weighted sharp function estimate and boundedness for commutator associated with singular integral operator with general kernel. N. Z. J. Math. 42, 17–26 (2012)

    MathSciNet  MATH  Google Scholar 

  27. Lu, S., Ding, Y., Yan, D.: Singular Integrals and Related Topics. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2007)

    Book  MATH  Google Scholar 

  28. Meyers, N.G.: Mean oscillation over cubes and Hölder continuity. Proc. Am. Math. Soc. 15, 717–721 (1994)

    MATH  Google Scholar 

  29. Morrey, C.B.: On the solutions of quasi-linear elliptic partial differential equations. Trans. Am. Math. Soc. 43(1), 126–166 (1938)

    Article  MathSciNet  MATH  Google Scholar 

  30. Ortiz-Caraballo, C.: Quadratic \(A_1\) bounds for commutators of singular integrals with \(BMO\) functions. Indiana Univ. Math. J. 60, 2107–2129 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. Paluszynski, M.: Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss. Indiana Univ. Math. J. 44, 1–17 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  32. Pérez, C.: Endpoint estimates for commutators of singular integral operators. J. Funct. Anal. 128, 163–185 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  33. Pérez, C., Torres, R.H.: Sharp maximal function estimates for multilinear singular integrals. Contemp. Math. 320, 323–331 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  34. Uchiyama, A.: On the compactness of operators of Hankel type. Tohoku Math. J. 30, 163–171 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  35. Wang, D., Zhou, J., Teng, Z.: Characterizations of the \(BMO\) and Lipschitz spaces via commutators on weak Lebesgue and Morrey spaces. arXiv: 1612.08819v1 (2016)

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Acknowledgements

The authors thank the referee cordially for his/her valuable remarks and suggestions, which make this article more readable.

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Authors and Affiliations

  1. School of Science, Jimei University, Xiamen, 361021, People’s Republic of China

    Weichao Guo

  2. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, People’s Republic of China

    Jiali Lian & Huoxiong Wu

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  1. Weichao Guo
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  2. Jiali Lian
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  3. Huoxiong Wu
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Correspondence to Huoxiong Wu.

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Supported by the NNSF of China (Nos. 11771358, 11871101, 11701112, 11671414).

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Guo, W., Lian, J. & Wu, H. The Unified Theory for the Necessity of Bounded Commutators and Applications. J Geom Anal 30, 3995–4035 (2020). https://doi.org/10.1007/s12220-019-00226-y

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  • DOI: https://doi.org/10.1007/s12220-019-00226-y

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