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Multiplier theorem for Hankel transform on Hardy spaces

  • Jacek DziubańskiEmail author
  • Marcin Preisner
Article

Abstract

The aim of this paper is to prove a multiplier theorem for the Hankel transform on the atomic Hardy space H 1(X), where X = ((0, ∞), x α dx) is the space of homogeneous type in the sense of Coifman–Weiss. The main tool is a maximal function characterization of H 1(X).

Keywords

Hankel transform Hardy spaces Multipliers 

Mathematics Subject Classification (2000)

42C15 42B30 42B15 42B35 

References

  1. 1.
    Betancor, J.J., Dziubański, J., Torrea, J.L.: On Hardy spaces associated with Bessel operators. J. Anal. Math. (2008, to appear)Google Scholar
  2. 2.
    Coifman R., Weiss G.: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83, 569–645 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G.: Higher Transcendental Functions, vol. 2. McGraw-Hill, New York (1953)Google Scholar
  4. 4.
    Garrigós G., Seeger A.: Characterizations of Hankel multipliers. Math. Ann. 342, 31–68 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Gasper G., Trebels W.: Multiplier criteria of Hörmander type for Fourier series and applications to Jacobi series and Hankel transforms. Math. Ann. 242(3), 225–240 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Gasper G., Trebels W.: A characterization of localized Bessel potential spaces and applications to Jacobi and Hankel multipliers. Studia Math. 65, 243–278 (1979)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Gosselin J., Stempak K.: A weak-type estimate for Fourier–Bessel multipliers. Proc. Am. Math. Soc. 106, 655–662 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Hörmander L.: Estimates for translation invariant operators in L p spaces. Acta. Math. 104, 93–140 (1960)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kapelko R.: A multiplier theorem for the Hankel transform. Rev. Math. Comput. 11, 281–288 (1998)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Macaulay-Owen P.: Parseval’s theorem for Hankel transforms. Proc. London Math. Soc. 45, 458–474 (1939)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Muckenhoupt B., Stein E.: Classical expansions and their relation to conjugate harmonic functions. Trans. Am. Math. Soc. 118, 17–92 (1965)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Stempak, K.: The Littlewood-Paley theory for the Fourier–Bessel transform, preprint no. 45, Math. Inst. Univ. Wroclaw, Poland (1985)Google Scholar
  13. 13.
    Stempak, K.: La théorie de Littlewood-Paley pour la transformation de Fourier-Bessel, C. R. Acad. Sc. Paris, t. 303, Rérie I, no. 1 (1986)Google Scholar
  14. 14.
    Tichmarsh E.C.: Introduction to the Theory of Fourier Integrals. Clarendon Press, Oxford (1937)Google Scholar
  15. 15.
    Watson G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge (1966)zbMATHGoogle Scholar
  16. 16.
    Weinstein A.: Discontinuous integrals and generalized potential theory. Trans. Am. Math. Soc. 63, 342–354 (1948)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of WrocławWrocławPoland

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