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Weighted estimates for commutators on nonhomogeneous spaces

Abstract

Let be a Borel measure on which may be nondoubling. The only condition that must satisfy is for any cube with sides parallel to the coordinate axes and for some fixed with. This paper is to establish the weighted norm inequality for commutators of Calderón-Zygmund operators with functions by an estimate for a variant of the sharp maximal function in the context of the nonhomogeneous spaces.

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References

  1. 1.

    Chen W, Sawyer E: A note on commutators of fractional integrals withfunctions. Illinois Journal of Mathematics 2002,46(4):1287–1298.

    MATH  MathSciNet  Google Scholar 

  2. 2.

    García-Cuerva J, Martell JM: Weighted inequalities and vector-valued Calderón-Zygmund operators on non-homogeneous spaces. Publicacions Matemàtiques 2000,44(2):613–640.

    MATH  Article  Google Scholar 

  3. 3.

    García-Cuerva J, Martell JM: On the existence of principal values for the Cauchy integral on weighted Lebesgue spaces for non-doubling measures. The Journal of Fourier Analysis and Applications 2001,7(5):469–487. 10.1007/BF02511221

    MATH  MathSciNet  Article  Google Scholar 

  4. 4.

    García-Cuerva J, Martell JM: Two-weight norm inequalities for maximal operators and fractional integrals on non-homogeneous spaces. Indiana University Mathematics Journal 2001,50(3):1241–1280.

    MATH  MathSciNet  Article  Google Scholar 

  5. 5.

    Han YC: Weighted estimates for higher-order commutators of singular integral operators on non-homogeneous spaces. Journal of South China Normal University. Natural Science Edition 2005,2005(3):92–99.

    MATH  MathSciNet  Google Scholar 

  6. 6.

    Komori Y: Weighted estimates for operators generated by maximal functions on nonhomogeneous spaces. Georgian Mathematical Journal 2005,12(1):121–130.

    MATH  MathSciNet  Google Scholar 

  7. 7.

    Mateu J, Mattila P, Nicolau A, Orobitg J: BMO for nondoubling measures. Duke Mathematical Journal 2000,102(3):533–565. 10.1215/S0012-7094-00-10238-4

    MATH  MathSciNet  Article  Google Scholar 

  8. 8.

    Nazarov F, Treil S, Volberg A: Cauchy integral and Calderón-Zygmund operators on nonhomogeneous spaces. International Mathematics Research Notices 1997,1997(15):703–726. 10.1155/S1073792897000469

    MATH  MathSciNet  Article  Google Scholar 

  9. 9.

    Nazarov F, Treil S, Volberg A: Weak type estimates and Cotlar inequalities for Calderón-Zygmund operators on nonhomogeneous spaces. International Mathematics Research Notices 1998,1998(9):463–487. 10.1155/S1073792898000312

    MATH  MathSciNet  Article  Google Scholar 

  10. 10.

    Nazarov F, Treil S, Volberg A: Accretive system-theorems on nonhomogeneous spaces. Duke Mathematical Journal 2002,113(2):259–312. 10.1215/S0012-7094-02-11323-4

    MATH  MathSciNet  Article  Google Scholar 

  11. 11.

    Orobitg J, Pérez C: weights for nondoubling measures inand applications. Transactions of the American Mathematical Society 2002,354(5):2013–2033. 10.1090/S0002-9947-02-02922-7

    MATH  MathSciNet  Article  Google Scholar 

  12. 12.

    Pérez C: Endpoint estimates for commutators of singular integral operators. Journal of Functional Analysis 1995,128(1):163–185. 10.1006/jfan.1995.1027

    MATH  MathSciNet  Article  Google Scholar 

  13. 13.

    Tolsa X: Cotlar's inequality without the doubling condition and existence of principal values for the Cauchy integral of measures. Journal für die reine und angewandte Mathematik 1998, 502: 199–235.

    MATH  MathSciNet  Google Scholar 

  14. 14.

    Tolsa X: -boundedness of the Cauchy integral operator for continuous measures. Duke Mathematical Journal 1999,98(2):269–304. 10.1215/S0012-7094-99-09808-3

    MATH  MathSciNet  Article  Google Scholar 

  15. 15.

    Tolsa X: BMO,, and Calderón-Zygmund operators for non doubling measures. Mathematische Annalen 2001,319(1):89–149. 10.1007/PL00004432

    MATH  MathSciNet  Article  Google Scholar 

  16. 16.

    Tolsa X: Littlewood-Paley theory and the theorem with non-doubling measures. Advances in Mathematics 2001,164(1):57–116. 10.1006/aima.2001.2011

    MATH  MathSciNet  Article  Google Scholar 

  17. 17.

    Tolsa X: The spacefor nondoubling measures in terms of a grand maximal operator. Transactions of the American Mathematical Society 2003,355(1):315–348. 10.1090/S0002-9947-02-03131-8

    MATH  MathSciNet  Article  Google Scholar 

  18. 18.

    Verdera J: The fall of the doubling condition in Calderón-Zygmund theory. Publicacions Matemàtiques 2002, 2002: Vol. Extra, 275–292.

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Wengu Chen.

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Chen, W., Zhao, B. Weighted estimates for commutators on nonhomogeneous spaces. J Inequal Appl 2006, 89396 (2006). https://doi.org/10.1155/JIA/2006/89396

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Keywords

  • Weighted Norm
  • Maximal Function
  • Borel Measure
  • Weighted Estimate
  • Norm Inequality
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