Skip to content

Advertisement

Journal of Inequalities and Applications
Journal of Inequalities and Applications Cover Image
  • Research Article
  • Open Access

Weighted estimates for commutators on nonhomogeneous spaces

Journal of Inequalities and Applications20062006:89396

https://doi.org/10.1155/JIA/2006/89396

©  W. Chen and B. Zhao. 2006

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

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.

Keywords

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

[123456789101112131415161718]

Authors’ Affiliations

(1)
Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing, 100088, China
(2)
Beijing October First School, Beijing, 100039, China

References

  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

Copyright

© W. Chen and B. Zhao. 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Advertisement