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The discrete maximal operator in metric spaces

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Abstract

We study the action of the so-called discrete maximal operator on Newtonian, Hölder and BMO spaces on metric measure spaces equipped with a doubling measure and a Poincaré inequality. The discrete maximal operator has better regularity properties than the standard Hardy-Littlewood maximal operator and hence is a more flexible tool in this context.

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References

  1. D. Aalto and J. Kinnunen, Maximal functions in Sobolev spaces, in Sobolev Spaces in Mathematics I, Springer, Berlin, 2008, pp. 25–68.

    Google Scholar 

  2. J. M. Aldaz and J. Pérez Lázaro, Boundedness and unboundedness results for some maximal operators on functions of bounded variation, J. Math. Anal. Appl. 337 (2008), 130–143.

    Article  MATH  MathSciNet  Google Scholar 

  3. C. Bennett, R. A. DeVore and R. Sharpley, Weak L ∞ and BMO, Ann. of Math. (2) 113 (1981), 601–611.

    Article  MathSciNet  Google Scholar 

  4. C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, New York, 1988.

    MATH  Google Scholar 

  5. A. Björn and J. Björn, Nonlinear potential theory on metric spaces, in preparation.

  6. S. M. Buckley, Inequalities of John-Nirenberg type in doubling spaces, J. Analyse Math. 79 (1999), 215–240.

    Article  MATH  MathSciNet  Google Scholar 

  7. S.M. Buckley, Is the maximal function of a Lipschitz function continuous?, Ann. Acad. Sci. Fenn. Math. 24 (1999), 519–528.

    MathSciNet  Google Scholar 

  8. F. Chiarenza and M. Frasca, Morrey spaces and Hardy-Littlewood maximal function, Rend. Mat. Appl. (7) 7 (1987), 273–279.

    MATH  MathSciNet  Google Scholar 

  9. R. R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certain espaces homogènes, Lecture Notes in Math. 242, Springer-Verlag, Berlin, 1971.

    MATH  Google Scholar 

  10. R. R. Coifman and R. Rochberg, Another characterization of BMO, Proc. Amer. Math. Soc. 79 (1980), 249–254.

    MATH  MathSciNet  Google Scholar 

  11. X. T. Duong and L. Yan, New function spaces of BMO type, the John-Nirenberg inequality, interpolation, and applications, Comm. Pure Appl. Math. 58 (2005), 1375–1420.

    Article  MATH  MathSciNet  Google Scholar 

  12. A. Fiorenza and M. Krbec, On some fundamental properties of the maximal operator, in Function Spaces and Applications (D. E. Edmunds et al., eds.), Narosa Publishing House, 2000.

  13. A. Fiorenza and M. Krbec, On the domain and range of the maximal operator, Nagoya Math. J. 158 (2000), 43–61.

    MATH  MathSciNet  Google Scholar 

  14. P. Hajłasz and J. Onninen, On boundedness of maximal functions in Sobolev spaces, Ann. Acad. Sci. Fenn. Math. 29 (2004), 167–176.

    MATH  MathSciNet  Google Scholar 

  15. S. Keith and X. Zhong, The Poincaré inequality is an open ended condition, Ann. of Math. 167 (2008), 575–599.

    Article  MATH  MathSciNet  Google Scholar 

  16. T. Kilpeläinen, J. Kinnunen and O. Martio, Sobolev spaces with zero boundary values on metric spaces, Potential Anal. 12 (2000), 233–247.

    Article  MathSciNet  Google Scholar 

  17. J. Kinnunen, The Hardy-Littlewood maximal function of a Sobolev function, Israel J. Math. 100 (1997), 117–124.

    Article  MATH  MathSciNet  Google Scholar 

  18. J. Kinnunen and V. Latvala, Lebesgue points for Sobolev functions on metric spaces, Rev. Mat. Iberoamericana 18 (2002), 685–700.

    MATH  MathSciNet  Google Scholar 

  19. J. Kinnunen and P. Lindqvist, The derivative of the maximal function, J. Reine Angew.Math. 503 (1998), 161–167.

    MATH  MathSciNet  Google Scholar 

  20. J. Kinnunen and O. Martio, Hardy’s inequalities for Sobolev functions, Math. Res. Lett. 4 (1997), 489–500.

    MATH  MathSciNet  Google Scholar 

  21. J. Kinnunen and O. Martio, Maximal operator and superharmonicity, in Function Spaces, Differential Operators and Nonlinear Analysis (Pudasjärvi, 1999), Acad. Sci. Czech Repub., Prague, 2000, pp. 157–169.

    Google Scholar 

  22. J. Kinnunen and H. Tuominen, Pointwise behaviour of M 1,1 Sobolev functions, Math. Z. 257 (2007), 613–630.

    Article  MATH  MathSciNet  Google Scholar 

  23. S. Korry, Boundedness of Hardy-Littlewood maximal operator in the framework of Lizorkin-Triebel spaces, Rev. Mat. Complut. 15 (2002), 401–416.

    MATH  MathSciNet  Google Scholar 

  24. S. Korry, A class of bounded operators on Sobolev spaces, Arch.Math. (Basel) 82 (2004), 40–50.

    MATH  MathSciNet  Google Scholar 

  25. M. Kronz, Some function spaces on spaces of homogeneous type, Manuscripta Math. 106 (2001), 219–248.

    Article  MATH  MathSciNet  Google Scholar 

  26. H. Luiro, Continuity of the maximal operator in Sobolev spaces, Proc. Amer. Math. Soc. 135 (2007), 243–251.

    Article  MATH  MathSciNet  Google Scholar 

  27. R. A. Macías and C. Segovia, A decomposition into atoms of distributions on spaces of homogeneous type, Adv. in Math. 33 (1979), 271–309.

    Article  MATH  MathSciNet  Google Scholar 

  28. P. MacManus, Poincaré inequalities and Sobolev spaces, Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations (El Escorial, 2000), Publ. Mat. 2002, Vol. Extra, 181–197.

  29. J. Mateu, P. Mattila, A. Nicolau and J. Orobitg, BMO for nondoubling measures, Duke Math. J. 102 (2000), 533–565.

    Article  MATH  MathSciNet  Google Scholar 

  30. W. Ou, The natural maximal operator on BMO, Proc. Amer. Math. Soc. 129 (2001), 2919–2921.

    Article  MATH  MathSciNet  Google Scholar 

  31. N. Shanmugalingam, Newtonian spaces: an extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoamericana 16 (2000), 243–279.

    MATH  MathSciNet  Google Scholar 

  32. N. Shanmugalingam, Harmonic functions on metric spaces, Illinois J. Math. 45 (2001), 1021–1050.

    MATH  MathSciNet  Google Scholar 

  33. J.-O. Strömberg and A. Torchinsky, Weighted Hardy Spaces, Lecture Notes in Math. 1381, Springer-Verlag, Berlin, 1989.

    MATH  Google Scholar 

  34. I. Wik, A comparison of the integrability of f and Mf with that of f #, University of Umeå, Preprint No. 2 (1983).

  35. J. Xiao, Bounded functions of vanishing mean oscillation on compact metric spaces, J. Funct. Anal. 209 (2004), 444–467.

    Article  MathSciNet  Google Scholar 

  36. D. Yang and Y. Zhou, Some new characterizations on functions of bounded mean oscillation, preprint.

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Authors and Affiliations

  1. Department of Mathematics, University of Turku, FI-20014, Turku, Finland

    Daniel Aalto

  2. Institute of Mathematics, Helsinki University of Technology, P.O.Box 1100, FI-02015, Helsinki, Finland

    Juha Kinnunen

Authors
  1. Daniel Aalto
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  2. Juha Kinnunen
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Correspondence to Daniel Aalto.

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Aalto, D., Kinnunen, J. The discrete maximal operator in metric spaces. JAMA 111, 369–390 (2010). https://doi.org/10.1007/s11854-010-0022-3

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  • DOI: https://doi.org/10.1007/s11854-010-0022-3

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