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Some remarks on the Hardy-Littlewood maximal function on variable Lp spaces

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Abstract

We show that any pointwise multiplier for BMO(ℝn) generates a function p from the class (ℝn) of those functions for which the Hardy-Littlewood maximal operator is bounded on the variable Lp space. In particular, this gives a positive answer to Diening's conjecture saying that there are discontinuous functions which nevertheless belong to (ℝn).

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References

  1. Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, New York, 1988

  2. Capone, C., Cruz-Uribe, D., Fiorenza, A.: The fractional maximal operator on variable Lp spaces. Preprint is available from http://www.iam.na.cnr.it/rapporti/2004/RT281_04.pdf

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

    Google Scholar 

  4. Cruz-Uribe, D., Fiorenza, A., Neugebauer, C.J.: The maximal function on variable Lp spaces. Ann. Acad. Sci. Fenn. Math. 28, 223–238 (2003); 29, 247–249 (2004)

    Google Scholar 

  5. Diening, L.: Maximal function on generalized Lebesgue spaces Lp(·). Math. Inequal. Appl. 7(2), 245–253 (2004)

    Google Scholar 

  6. Diening, L.: Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces. Fakultät für Mathematik und Physik, Albert-Ludwigs-Universität Freiburg. Preprint no. 21, 2003

  7. García-Cuerva, J., Rubio de Francia, J.: Weighted norm inequalities and related topics. North-Holland Math. Studies 116, North-Holland, Amsterdam, 1985

  8. Janson, S.: On functions with conditions on the mean oscillation. Ark. Mat. 14, 189–196 (1976)

    Google Scholar 

  9. Kováčik, O., Rákosník, J.: On spaces Lp(x) and Wk,p(x). Czechoslovak Math. J. 41(116)(4), 592–618 (1991)

    Google Scholar 

  10. Musielak, J.: Orlicz Spaces and Modular Spaces. Lecture Notes in Mathematics 1034, Springer-Verlag, Berlin, 1983

  11. Nakai, E., Yabuta, K.: Pointwise multipliers for functions of bounded mean oscillation. J. Math. Soc. Japan 37, 207–218 (1985)

    Google Scholar 

  12. Nekvinda, A.: Hardy-Littlewood maximal operator on Lp(x)(ℝn). Math. Inequal. Appl. 7(2), 255–265 (2004)

    Google Scholar 

  13. Nekvinda, A.: A note on maximal operator on and on Lp(x)(ℝ). Faculty of Civil Engeneering, Czech Technical University in Prague. Preprint no. 1, 2004

  14. Pick, L., Růžička, M.: An example of a space of Lp(x) on which the Hardy-Littlewood maximal operator is not bounded. Expo. Math. 19(4), 369–371 (2001)

    Article  Google Scholar 

  15. Spanne, S.: Some function spaces defined using the mean oscillation over cubes. Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat. III. 19, 593–608 (1965)

    Google Scholar 

  16. Stegenga, D.A.: Bounded Toeplitz operators on H1 and applications of the duality between H1 and the functions of bounded mean oscillation. Am. J. Math. 98, 573–589 (1976)

    Google Scholar 

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  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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Lerner, A. Some remarks on the Hardy-Littlewood maximal function on variable Lp spaces. Math. Z. 251, 509–521 (2005). https://doi.org/10.1007/s00209-005-0818-5

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  • DOI: https://doi.org/10.1007/s00209-005-0818-5

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