Open Access

Uniform Boundedness for Approximations of the Identity with Nondoubling Measures

Journal of Inequalities and Applications20072007:019574

DOI: 10.1155/2007/19574

Received: 15 May 2007

Accepted: 19 August 2007

Published: 16 October 2007

Abstract

Let https://static-content.springer.com/image/art%3A10.1155%2F2007%2F19574/MediaObjects/13660_2007_Article_1678_IEq1_HTML.gif be a nonnegative Radon measure on https://static-content.springer.com/image/art%3A10.1155%2F2007%2F19574/MediaObjects/13660_2007_Article_1678_IEq2_HTML.gif which satisfies the growth condition that there exist constants https://static-content.springer.com/image/art%3A10.1155%2F2007%2F19574/MediaObjects/13660_2007_Article_1678_IEq3_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2007%2F19574/MediaObjects/13660_2007_Article_1678_IEq4_HTML.gif such that for all https://static-content.springer.com/image/art%3A10.1155%2F2007%2F19574/MediaObjects/13660_2007_Article_1678_IEq5_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2007%2F19574/MediaObjects/13660_2007_Article_1678_IEq6_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2007%2F19574/MediaObjects/13660_2007_Article_1678_IEq7_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2007%2F19574/MediaObjects/13660_2007_Article_1678_IEq8_HTML.gif is the open ball centered at https://static-content.springer.com/image/art%3A10.1155%2F2007%2F19574/MediaObjects/13660_2007_Article_1678_IEq9_HTML.gif and having radius https://static-content.springer.com/image/art%3A10.1155%2F2007%2F19574/MediaObjects/13660_2007_Article_1678_IEq10_HTML.gif . In this paper, the authors establish the uniform boundedness for approximations of the identity introduced by Tolsa in the Hardy space https://static-content.springer.com/image/art%3A10.1155%2F2007%2F19574/MediaObjects/13660_2007_Article_1678_IEq11_HTML.gif and the BLO-type space RBLO https://static-content.springer.com/image/art%3A10.1155%2F2007%2F19574/MediaObjects/13660_2007_Article_1678_IEq12_HTML.gif . Moreover, the authors also introduce maximal operators https://static-content.springer.com/image/art%3A10.1155%2F2007%2F19574/MediaObjects/13660_2007_Article_1678_IEq13_HTML.gif (homogeneous) and https://static-content.springer.com/image/art%3A10.1155%2F2007%2F19574/MediaObjects/13660_2007_Article_1678_IEq14_HTML.gif (inhomogeneous) associated with a given approximation of the identity https://static-content.springer.com/image/art%3A10.1155%2F2007%2F19574/MediaObjects/13660_2007_Article_1678_IEq15_HTML.gif , and prove that https://static-content.springer.com/image/art%3A10.1155%2F2007%2F19574/MediaObjects/13660_2007_Article_1678_IEq16_HTML.gif is bounded from https://static-content.springer.com/image/art%3A10.1155%2F2007%2F19574/MediaObjects/13660_2007_Article_1678_IEq17_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2007%2F19574/MediaObjects/13660_2007_Article_1678_IEq18_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2007%2F19574/MediaObjects/13660_2007_Article_1678_IEq19_HTML.gif is bounded from the local atomic Hardy space https://static-content.springer.com/image/art%3A10.1155%2F2007%2F19574/MediaObjects/13660_2007_Article_1678_IEq20_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2007%2F19574/MediaObjects/13660_2007_Article_1678_IEq21_HTML.gif . These results are proved to play key roles in establishing relations between https://static-content.springer.com/image/art%3A10.1155%2F2007%2F19574/MediaObjects/13660_2007_Article_1678_IEq22_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2007%2F19574/MediaObjects/13660_2007_Article_1678_IEq23_HTML.gif , BMO-type spaces RBMO https://static-content.springer.com/image/art%3A10.1155%2F2007%2F19574/MediaObjects/13660_2007_Article_1678_IEq24_HTML.gif and rbmo https://static-content.springer.com/image/art%3A10.1155%2F2007%2F19574/MediaObjects/13660_2007_Article_1678_IEq25_HTML.gif as well as RBLO https://static-content.springer.com/image/art%3A10.1155%2F2007%2F19574/MediaObjects/13660_2007_Article_1678_IEq26_HTML.gif and rblo https://static-content.springer.com/image/art%3A10.1155%2F2007%2F19574/MediaObjects/13660_2007_Article_1678_IEq27_HTML.gif , and also in characterizing rbmo https://static-content.springer.com/image/art%3A10.1155%2F2007%2F19574/MediaObjects/13660_2007_Article_1678_IEq28_HTML.gif and rblo https://static-content.springer.com/image/art%3A10.1155%2F2007%2F19574/MediaObjects/13660_2007_Article_1678_IEq29_HTML.gif .

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Authors’ Affiliations

(1)
School of Mathematical Sciences, Beijing Normal University

References

  1. Jiang Y: Spaces of type BLO for non-doubling measures. Proceedings of the American Mathematical Society 2005,133(7):2101–2107. 10.1090/S0002-9939-05-07795-6MathSciNetView ArticleMATHGoogle Scholar
  2. Mateu J, Mattila P, Nicolau A, Orobitg J: BMO for nondoubling measures. Duke Mathematical Journal 2000,102(3):533–565. 10.1215/S0012-7094-00-10238-4MathSciNetView ArticleMATHGoogle Scholar
  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/S1073792897000469MathSciNetView ArticleMATHGoogle Scholar
  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/S1073792898000312MathSciNetView ArticleMATHGoogle Scholar
  5. Nazarov F, Treil S, Volberg A: Accretive system-theorems on nonhomogeneous spaces. Duke Mathematical Journal 2002,113(2):259–312. 10.1215/S0012-7094-02-11323-4MathSciNetView ArticleMATHGoogle Scholar
  6. Nazarov F, Treil S, Volberg A: The-theorem on non-homogeneous spaces. Acta Mathematica 2003,190(2):151–239. 10.1007/BF02392690MathSciNetView ArticleMATHGoogle Scholar
  7. Tolsa X: BMO,, and Calderón-Zygmund operators for non doubling measures. Mathematische Annalen 2001,319(1):89–149. 10.1007/PL00004432MathSciNetView ArticleMATHGoogle Scholar
  8. Tolsa X: Littlewood-Paley theory and thetheorem with non-doubling measures. Advances in Mathematics 2001,164(1):57–116. 10.1006/aima.2001.2011MathSciNetView ArticleMATHGoogle Scholar
  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/S0002-9947-02-03131-8MathSciNetView ArticleMATHGoogle Scholar
  10. Tolsa X: Painlevé's problem and the semiadditivity of analytic capacity. Acta Mathematica 2003,190(1):105–149. 10.1007/BF02393237MathSciNetView ArticleMATHGoogle Scholar
  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.2004.0021MathSciNetView ArticleMATHGoogle Scholar
  12. Tolsa X: Bilipschitz maps, analytic capacity, and the Cauchy integral. Annals of Mathematics. Second Series 2005,162(3):1243–1304. 10.4007/annals.2005.162.1243MathSciNetView ArticleMATHGoogle Scholar
  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. Tolsa X: Painlevé's problem and analytic capacity. Collectanea Mathematica 2006, Extra: 89–125.MathSciNetGoogle Scholar
  15. Verdera J: The fall of the doubling condition in Calderón-Zygmund theory. Publicacions Matemàtiques 2002, Extra: 275–292.MathSciNetView ArticleGoogle Scholar
  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.View ArticleGoogle Scholar
  17. Hu G, Yang D, Yang D: , bmo, blo and Littlewood-Paley g-functions with non-doubling measures. submitted submitted
  18. Yang D, Yang D: Endpoint estimates for homogeneous Littlewood-Paley-functions with non-doubling measures. submitted submitted
  19. Goldberg D: A local version of real Hardy spaces. Duke Mathematical Journal 1979,46(1):27–42. 10.1215/S0012-7094-79-04603-9MathSciNetView ArticleMATHGoogle Scholar
  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/S0002-9939-05-07781-6MathSciNetView ArticleMATHGoogle Scholar

Copyright

© D. Yang and D. Yang 2007

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.