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Maximal functions associated with Fourier multipliers of Mikhlin-Hörmander type

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Abstract.

We show that maximal operators formed by dilations of Mikhlin- Hörmander multipliers are typically not bounded on Lp(ℝd). We also give rather weak conditions in terms of the decay of such multipliers under which Lp boundedness of the maximal operators holds.

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References

  1. Bergh, J., Löfström, J.: Interpolation spaces. Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976

  2. Carbery, A.: Radial Fourier multipliers and associated maximal functions. Recent progress in Fourier analysis, I. Peral, J.L. Rubio de Francia (eds.), North Holland, 1985

  3. Connett, W.C., Schwartz, A.L.: A remark about Calderón’s upper s method of interpolation. Interpolation and allied topics in analysis (Lund, 1983), Lecture Notes in Math., 1070, Springer, Berlin, 1984, pp. 48–53

  4. Dappa, H., Trebels, W.: On maximal functions generated by Fourier multipliers. Ark. Mat. 23, 241–259 (1985)

    MathSciNet  MATH  Google Scholar 

  5. Fefferman, C., Stein, E.M.: Hp spaces of several variables. Acta Math. 129, 137–193 (1972)

    MATH  Google Scholar 

  6. Grafakos, L.: Classical and Modern Fourier Analysis. Prentice Hall, Upper Saddle River NJ, 2003

  7. Hörmander, L.: Estimates for translation invariant operators in Lp spaces. Acta Math. 104, 93–139 (1960)

    Google Scholar 

  8. Mikhlin, S.G.: On the multipliers of Fourier integrals. (Russian) Dokl. Akad. Nauk SSSR (N.S.) 109, 701–703 (1956)

    MATH  Google Scholar 

  9. Rubio de Francia, J.-L.: Maximal functions and Fourier transforms. Duke Math. J. 53, 395–404 (1986)

    MATH  Google Scholar 

  10. Seeger, A.: Estimates near L1 for Fourier multipliers and maximal functions. Arch. Math. (Basel) 53(2), 188–193 (1989)

    MATH  Google Scholar 

  11. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press, Princeton, NJ, 1971

  12. Wolff, T.H.: A note on interpolation spaces. Harmonic Analysis (Minneapolis 1981), 199-204, Lecture Notes in Mathematics, 908, Springer, Berlin-New York, 1982

  13. Zo, F.: A note on approximation of the identity. Studia Math 55, 111–122 (1976)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of California, Berkeley, CA, 94720-3840, USA

    Michael Christ

  2. Department of Mathematics, University of Missouri, Columbia, MO, 65211, USA

    Loukas Grafakos & Petr Honzík

  3. Department of Mathematics, University of Wisconsin, Madison, WI, 53706, USA

    Andreas Seeger

Authors
  1. Michael Christ
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  2. Loukas Grafakos
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  3. Petr Honzík
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  4. Andreas Seeger
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Corresponding author

Correspondence to Loukas Grafakos.

Additional information

Christ, Grafakos and Seeger were supported in part by NSF grants. Honzík was supported by 201/03/0931 Grant Agency of the Czech Republic

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Christ, M., Grafakos, L., Honzík, P. et al. Maximal functions associated with Fourier multipliers of Mikhlin-Hörmander type. Math. Z. 249, 223–240 (2005). https://doi.org/10.1007/s00209-004-0698-0

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  • DOI: https://doi.org/10.1007/s00209-004-0698-0

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