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Some remarks on the Fefferman-Stein inequality

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Abstract

We investigate the Fefferman-Stein inequality related to a function f and the sharp maximal function f # on a Banach function space X. It is proved that this inequality is equivalent to a certain boundedness property of the Hardy-Littlewood maximal operator M. The latter property is shown to be self-improving. We apply our results in several directions. First, we show the existence of nontrivial spaces X for which the lower operator norm of M is equal to 1. Second, in the case when X is the weighted Lebesgue space L p(w), we obtain a new approach to the results of Sawyer and Yabuta concerning the C p condition.

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References

  1. J. Alvarez and C. Pérez, Estimates with A ∞ weights for various singular integral operators, Boll. Un. Mat. Ital. A)(7) 8 (1994), 123–133.

    MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  3. R. R. Coifman, Distribution function inequalities for singular integrals, Proc. Nat. Acad. Sci. U.S.A. 69 (1972), 2838–2839.

    Article  MATH  MathSciNet  Google Scholar 

  4. R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia. Math. 15 (1974), 241–250.

    MathSciNet  Google Scholar 

  5. G. P. Curbera, J. García-Cuerva, J. M. Martell and C. Pérez, Extrapolation with weights, rearrangement-invariant function spaces, modular inequalities and applications to singular integrals, Adv. Math. 203 (2006), 256–318.

    Article  MATH  MathSciNet  Google Scholar 

  6. C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107–115.

    Article  MATH  MathSciNet  Google Scholar 

  7. C. Fefferman and E.M. Stein, H p spaces of several variables, Acta Math. 129 (1972), 137–193.

    Article  MATH  MathSciNet  Google Scholar 

  8. L. Grafakos, Classical and Modern Fourier Analysis, Prentice Hall, Engelwood Cliffs, NJ, 2004.

    MATH  Google Scholar 

  9. P. Janakiraman, Limiting weak-type behavior for singular integral and maximal operators, Trans. Amer. Math. Soc. 358 (2006), 1937–1952.

    Article  MATH  MathSciNet  Google Scholar 

  10. B. Jawerth and A. Torchinsky, Local sharp maximal functions, J. Approx. Theory 43 (1985), 231–270.

    Article  MATH  MathSciNet  Google Scholar 

  11. F. John, Quasi-isometric Mappings, Seminari 1962–1963 di Analisi, Algebra, Geometria e Topologia, Rome, 1965.

    Google Scholar 

  12. L. Kahanpää and L. Mejlbro, Some new results on the Muckenhoupt conjecture concerning weighted norm inequalities connecting the Hilbert transform with the maximal function, in Proceedings of the second Finnish-Polish Summer School in Complex Analysis (Jyväskylä, 1983), Bericht, 28, Univ. Jyväskylä, Jyväskylä, 1984, pp. 53–72.

    Google Scholar 

  13. S. Korry, Fixed points of the Hardy-Littlewood maximal operator, Collect. Math. 52 (2001), 289–294.

    MATH  MathSciNet  Google Scholar 

  14. A. K. Lerner, On the John-Strömberg characterization of BMO for nondoubling measures, Real Anal. Exchange 28 (2003), 649–660.

    MATH  MathSciNet  Google Scholar 

  15. A. K. Lerner, Weighted norm inequalities for the local sharp maximal function, J. Fourier Anal. Appl. 10 (2004), 465–474.

    Article  MATH  MathSciNet  Google Scholar 

  16. A. K. Lerner, Weighted rearrangement inequalities for local sharp maximal functions, Trans. Amer. Math. Soc. 357 (2005), 2445–2465.

    Article  MATH  MathSciNet  Google Scholar 

  17. A. K. Lerner, BMO-boundedness of the maximal operator for arbitrary measures, Israel J. Math. 159 (2007), 243–252.

    Article  MATH  MathSciNet  Google Scholar 

  18. A. K. Lerner and S. Ombrosi, A boundedness criterion for general maximal operators, Publ. Mat. 54 (2010), 53–71.

    MATH  MathSciNet  Google Scholar 

  19. A. K. Lerner, S. Ombrosi, C. Pérez, R. H. Torres and R. Trujillo-Gonzalez, New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory, Adv.Math. 220 (2009), 1222–1264.

    Article  MATH  MathSciNet  Google Scholar 

  20. A. K. Lerner and C. Pérez, A new characterization of the Muckenhoupt Ap weights through an extension of the Lorentz-Shimogaki theorem, Indiana Univ. Math. J. 56 (2007), 2697–2722.

    Article  MATH  MathSciNet  Google Scholar 

  21. J. Martín and J. Soria, Characterization of rearrangement invariant spaces with fixed points for the Hardy-Littlewood maximal operator, Ann. Acad. Sci. Fenn. Math. 31 (2006), 39–46.

    MATH  MathSciNet  Google Scholar 

  22. B. Muckenhoupt, Norm inequalities relating the Hilbert ransform to the Hardy-Littlewood maximal function, in Functional Analysis and Approximation (Oberwolfach, 1980), Birkhäuser, Basel-Boston, Mass., 1981, pp. 219–231.

    Google Scholar 

  23. J. L. Rubio de Francia, Factorization theory and A p weights, Amer. J. Math. 106 (1984), 533–547.

    Article  MATH  MathSciNet  Google Scholar 

  24. E. T. Sawyer, Two weight norm inequalities for certain maximal and integral operators, in Harmonic Analysis (Minneapolis, Minn., 1981), Lecture Notes in Math. 908, Springer, Berlin-New York, 1982, pp. 102–127.

    Google Scholar 

  25. E. T. Sawyer, Norm inequalities relating singular integrals and the maximal function, Studia Math. 75 (1983), 253–263.

    MATH  MathSciNet  Google Scholar 

  26. J.-O. Strömberg, Bounded mean oscillation with Orlicz norms and duality of Hardy spaces, Indiana Univ. Math. J. 28 (1979), 511–544.

    Article  MathSciNet  Google Scholar 

  27. A. de la Torre, On the adjoint of the maximal function, in Function Spaces, Differential Operators and Nonlinear Analysis (Paseky nad Jizerou, 1995), Prometheus, Prague, 1996, pp. 189–194.

  28. K. Yabuta, Sharp maximal function and Cp condition, Arch. Math. 55 (1990), 151–155.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Bar-Ilan University, 52900, Ramat Gan, Israel

    Andrei K. Lerner

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  1. Andrei K. Lerner
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Correspondence to Andrei K. Lerner.

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Lerner, A.K. Some remarks on the Fefferman-Stein inequality. JAMA 112, 329–349 (2010). https://doi.org/10.1007/s11854-010-0032-1

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  • DOI: https://doi.org/10.1007/s11854-010-0032-1

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