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  • Research Article
  • Open Access

Weighted estimates for commutators on nonhomogeneous spaces

Journal of Inequalities and Applications20062006:89396

  • Received: 14 November 2005
  • Accepted: 11 March 2006
  • Published:


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.


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


Authors’ Affiliations

Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing, 100088, China
Beijing October First School, Beijing, 100039, China


  1. Chen W, Sawyer E: A note on commutators of fractional integrals with functions. Illinois Journal of Mathematics 2002,46(4):1287–1298.MATHMathSciNetGoogle Scholar
  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.MATHView ArticleGoogle Scholar
  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/BF02511221MATHMathSciNetView ArticleGoogle Scholar
  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.MATHMathSciNetView ArticleGoogle Scholar
  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.MATHMathSciNetGoogle Scholar
  6. Komori Y: Weighted estimates for operators generated by maximal functions on nonhomogeneous spaces. Georgian Mathematical Journal 2005,12(1):121–130.MATHMathSciNetGoogle Scholar
  7. Mateu J, Mattila P, Nicolau A, Orobitg J: BMO for nondoubling measures. Duke Mathematical Journal 2000,102(3):533–565. 10.1215/S0012-7094-00-10238-4MATHMathSciNetView ArticleGoogle Scholar
  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/S1073792897000469MATHMathSciNetView ArticleGoogle Scholar
  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/S1073792898000312MATHMathSciNetView ArticleGoogle Scholar
  10. Nazarov F, Treil S, Volberg A: Accretive system -theorems on nonhomogeneous spaces. Duke Mathematical Journal 2002,113(2):259–312. 10.1215/S0012-7094-02-11323-4MATHMathSciNetView ArticleGoogle Scholar
  11. Orobitg J, Pérez C: weights for nondoubling measures inand applications . Transactions of the American Mathematical Society 2002,354(5):2013–2033. 10.1090/S0002-9947-02-02922-7MATHMathSciNetView ArticleGoogle Scholar
  12. Pérez C: Endpoint estimates for commutators of singular integral operators. Journal of Functional Analysis 1995,128(1):163–185. 10.1006/jfan.1995.1027MATHMathSciNetView ArticleGoogle Scholar
  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.MATHMathSciNetGoogle Scholar
  14. Tolsa X: -boundedness of the Cauchy integral operator for continuous measures. Duke Mathematical Journal 1999,98(2):269–304. 10.1215/S0012-7094-99-09808-3MATHMathSciNetView ArticleGoogle Scholar
  15. Tolsa X: BMO, , and Calderón-Zygmund operators for non doubling measures. Mathematische Annalen 2001,319(1):89–149. 10.1007/PL00004432MATHMathSciNetView ArticleGoogle Scholar
  16. Tolsa X: Littlewood-Paley theory and the theorem with non-doubling measures. Advances in Mathematics 2001,164(1):57–116. 10.1006/aima.2001.2011MATHMathSciNetView ArticleGoogle Scholar
  17. Tolsa X: The space for nondoubling measures in terms of a grand maximal operator. Transactions of the American Mathematical Society 2003,355(1):315–348. 10.1090/S0002-9947-02-03131-8MATHMathSciNetView ArticleGoogle Scholar
  18. Verdera J: The fall of the doubling condition in Calderón-Zygmund theory. Publicacions Matemàtiques 2002, 2002: Vol. Extra, 275–292.MathSciNetView ArticleMATHGoogle Scholar