Mathematische Zeitschrift

, Volume 265, Issue 2, pp 417–435 | Cite as

New results on restriction of Fourier multipliers

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Abstract

We develop an extension of the Transference methods introduced by R. Coifman and G. Weiss and apply it to study the problem of the restriction of Fourier multipliers between rearrangement invariant spaces, obtaining natural extensions of the classical de Leeuw’s result and its further extension to maximal Fourier multipliers due to C. Kenig and P. Tomas.

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References

  1. 1.
    Asmar N., Berkson E., Gillespie T.A.: Transference of strong type maximal inequalities by separation-preserving representations. Am. J. Math. 113(1), 47–49 (1991)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Asmar N., Berkson E., Gillespie T.A.: Transfert des opérateurs maximaux par des représentations conservant la séparation. C. R. Acad. Sci. Paris Sér. I Math. 309(3), 163–166 (1989)MATHMathSciNetGoogle Scholar
  3. 3.
    Asmar N., Berkson E., Gillespie T.A.: Transfert des multiplicateurs de type faible. C. R. Acad. Sci. Paris Sér. I Math. 311(3), 173–176 (1990)MATHMathSciNetGoogle Scholar
  4. 4.
    Bennett C., Sharpley R.: Interpolation of operators. Pure and Applied Mathematics, vol. 129. Academic Press Inc., Boston (1988)Google Scholar
  5. 5.
    Berkson E., Blasco O., Carro M.J., Gillespie T.A.: Discretization and transference of bisublinear maximal operators. J. Fourier Anal. Appl. 12(4), 447–481 (2006)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bernardis A.L., Martín-Reyes F.J.: Singular integrals in the Cesàro sense. J. Fourier Anal. Appl. 6(2), 143–152 (2000)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Blasco O., Carro M., Gillespie T.A.: Bilinear Hilbert transform on measure spaces. J. Fourier Anal. Appl. 11(4), 459–470 (2005)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Blasco O., Villarroya F.: Transference of bilinear multiplier operators on Lorentz spaces. Ill. J. Math. 47(4), 1327–1343 (2003)MATHMathSciNetGoogle Scholar
  9. 9.
    Carro M., Soria J.: Transference theory on Hardy and Sobolev spaces. Colloq. Math. 74(1), 47–69 (1997)MATHMathSciNetGoogle Scholar
  10. 10.
    Carro M.J., Raposo J.A., Soria J.: Recent developments in the theory of Lorentz spaces and weighted inequalities. Mem. Am. Math. Soc. 187(877), xii+128 (2007)MathSciNetGoogle Scholar
  11. 11.
    Carro M.J., Soria J.: Weighted Lorentz spaces and the Hardy operator. J. Funct. Anal. 112(2), 480–494 (1993)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Chandarana S.: L p-bounds for hypersingular integral operators along curves. Pacific J. Math. 175, 389–416 (1996)MathSciNetGoogle Scholar
  13. 13.
    Coifman, R.R., Weiss, G.: Transference Methods in Analysis. American Mathematical Society, Providence (1976) (Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 31)Google Scholar
  14. 14.
    Cowling M.G., Fournier J.J.F.: Inclusions and noninclusion of spaces of convolution operators. Trans. Am. Math. Soc. 221(1), 59–95 (1976)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Fefferman C.: Inequalities for strongly singular convolution operators. Acta Math. 124, 9–36 (1970)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Feichtinger, H.G.: Banach convolution algebras of Wiener type. In: Functions, series, operators, vol. I, II (Budapest, 1980), Colloq. Math. Soc. János Bolyai, vol. 35, pp. 509–524. North-Holland, Amsterdam (1983)Google Scholar
  17. 17.
    Feichtinger H.G., Luef F.: Wiener amalgam spaces for the fundamental identity of gabor analysis. Collect. Math. Vol. Extra(1), 233–253 (2006)MathSciNetGoogle Scholar
  18. 18.
    Hardy G.H.: A theorem concerning Taylor’s series. Quart. J. Pure Appl. Math. 44, 147–160 (1913)MATHGoogle Scholar
  19. 19.
    Hirschman I.I. Jr.: On multiplier transformations. Duke Math. J 26, 221–242 (1959)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Kenig C.E., Tomas P.A.: Maximal operators defined by Fourier multipliers. Studia Math. 68(1), 79–83 (1980)MATHMathSciNetGoogle Scholar
  21. 21.
    Laghi N., Lyall N.: Strongly singular integrals along curves. Pacific J. Math. 233(2), 403–415 (2007)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    de Leeuw K.: On L p multipliers. Ann. Math. 81(2), 364–379 (1965)CrossRefGoogle Scholar
  23. 23.
    Raposo J.A.: Weak type (1,1) multipliers on LCA groups. Studia Math. 122(2), 123–130 (1997)MATHMathSciNetGoogle Scholar
  24. 24.
    Seeger A., Tao T.: Sharp Lorentz space estimates for rough operators. Math. Ann. 320(2), 381–415 (2001)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Stein, E.M.: Singular integrals, harmonic functions, and differentiability properties of functions of several variables. In: Singular integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), pp. 316–335. Amer. Math. Soc., Providence (1967)Google Scholar
  26. 26.
    Wainger S.: Special trigonometric series in k-dimensions. Mem. Am. Math. Soc. No. 59, 102 (1965)MathSciNetGoogle Scholar

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© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Departament de Matemàtica Aplicada i AnàlisiUniversitat de BarcelonaBarcelonaSpain

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