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Uniform Boundedness for Approximations of the Identity with Nondoubling Measures

Abstract

Let be a nonnegative Radon measure on which satisfies the growth condition that there exist constants and such that for all and,, where is the open ball centered at and having radius. In this paper, the authors establish the uniform boundedness for approximations of the identity introduced by Tolsa in the Hardy space and the BLO-type space RBLO. Moreover, the authors also introduce maximal operators (homogeneous) and (inhomogeneous) associated with a given approximation of the identity, and prove that is bounded from to and is bounded from the local atomic Hardy space to. These results are proved to play key roles in establishing relations between and, BMO-type spaces RBMO and rbmo as well as RBLO and rblo, and also in characterizing rbmo and rblo.

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References

  1. 1.

    Jiang Y: Spaces of type BLO for non-doubling measures. Proceedings of the American Mathematical Society 2005,133(7):2101–2107. 10.1090/S0002-9939-05-07795-6

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    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

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    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

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    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

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    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

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Nazarov F, Treil S, Volberg A: The-theorem on non-homogeneous spaces. Acta Mathematica 2003,190(2):151–239. 10.1007/BF02392690

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

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

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

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

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    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

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Tolsa X: Painlevé's problem and the semiadditivity of analytic capacity. Acta Mathematica 2003,190(1):105–149. 10.1007/BF02393237

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Tolsa X: The semiadditivity of continuous analytic capacity and the inner boundary conjecture. American Journal of Mathematics 2004,126(3):523–567. 10.1353/ajm.2004.0021

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Tolsa X: Bilipschitz maps, analytic capacity, and the Cauchy integral. Annals of Mathematics. Second Series 2005,162(3):1243–1304. 10.4007/annals.2005.162.1243

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Tolsa X: Analytic capacity and Calderón-Zygmund theory with non doubling measures. In Seminar of Mathematical Analysis, Colecc. Abierta. Volume 71. Universidad de Sevilla. Secretariado de Publicaciones, Sevilla, Spain; 2004:239–271.

    Google Scholar 

  14. 14.

    Tolsa X: Painlevé's problem and analytic capacity. Collectanea Mathematica 2006, Extra: 89–125.

    MathSciNet  Google Scholar 

  15. 15.

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

    MathSciNet  Article  Google Scholar 

  16. 16.

    Volberg A: Calderón-Zygmund Capacities and Operators on Nonhomogeneous Spaces, CBMS Regional Conference Series in Mathematics. Volume 100. American Mathematical Society Providence, RI, USA; 2003:iv+167.

    Google Scholar 

  17. 17.

    Hu G, Yang D, Yang D: , bmo, blo and Littlewood-Paley g-functions with non-doubling measures. submitted submitted

  18. 18.

    Yang D, Yang D: Endpoint estimates for homogeneous Littlewood-Paley-functions with non-doubling measures. submitted submitted

  19. 19.

    Goldberg D: A local version of real Hardy spaces. Duke Mathematical Journal 1979,46(1):27–42. 10.1215/S0012-7094-79-04603-9

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Chen W, Meng Y, Yang D: Calderón-Zygmund operators on Hardy spaces without the doubling condition. Proceedings of the American Mathematical Society 2005,133(9):2671–2680. 10.1090/S0002-9939-05-07781-6

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Dachun Yang.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Yang, D., Yang, D. Uniform Boundedness for Approximations of the Identity with Nondoubling Measures. J Inequal Appl 2007, 019574 (2007). https://doi.org/10.1155/2007/19574

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Keywords

  • Growth Condition
  • Hardy Space
  • Maximal Operator
  • Open Ball
  • Radon Measure
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