Journal of Fourier Analysis and Applications

, Volume 19, Issue 2, pp 417–437 | Cite as

Multivariate Hörmander-Type Multiplier Theorem for the Hankel transform

  • Jacek Dziubański
  • Marcin Preisner
  • Błażej Wróbel


Let \(\mathcal{H}(f)(x)=\int_{(0,\infty)^{d}} f(\lambda) E_{x}(\lambda) d\nu(\lambda )\), be the multivariate Hankel transform, where \(E_{x}(\lambda)=\prod_{k=1}^{d} (x_{k} \lambda_{k})^{-\alpha _{k}+1/2}J_{\alpha_{k}-1/2}(x_{k} \lambda_{k})\), with (λ)=λ 2α , α=(α 1,…,α d ). We give sufficient conditions on a bounded function m(λ) which guarantee that the operator \(\mathcal{H}(m\mathcal{H} f)\) is bounded on L p () and of weak-type (1,1), or bounded on the Hardy space H 1((0,∞) d ,) in the sense of Coifman-Weiss.


Spectral multiplier Bessel operator Hankel transform Hardy space 

Mathematics Subject Classification

42B15 42B20 42B30 



The authors would like to thank Alessio Martini for discussions on spectral multipliers, Adam Nowak and Tomasz Z. Szarek for their useful remarks, Jacek Zienkiewicz for pointing out to us Example 5.1, and the referees for their helpful comments and suggestions.


  1. 1.
    Alexopoulos, G.: Spectral multipliers on Lie groups of polynomial growth. Proc. Am. Math. Soc. 120(3), 973–979 (1994) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bergh, J., Löfström, J.: Interpolation Spaces: an Introduction. Springer, Berlin (1976) MATHCrossRefGoogle Scholar
  3. 3.
    Betancor, J.J., Castro, A.J., Curbelo, J.: Spectral multipliers for multidimensional Bessel operators. J. Fourier Anal. Appl. 17(5), 932–975 (2011) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Betancor, J.J., Dziubański, J., Torrea, J.L.: On Hardy spaces associated with Bessel operators. J. Anal. Math. 107, 195–219 (2009) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Bloom, W.R., Xu, Z.: Fourier multipliers for L p on Chébli-Trimeche hypergroups. Proc. Lond. Math. Soc. 80(3), 643–664 (2000) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Christ, M.: L p bounds for spectral multipliers on nilpotent groups. Trans. Am. Math. Soc. 328(1), 73–81 (1991) MathSciNetMATHGoogle Scholar
  7. 7.
    Coifman, R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83(4), 569–645 (1977) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Duoandikoetxea, J.: Fourier Analysis. Am. Math. Soc., Providence (2001) MATHGoogle Scholar
  9. 9.
    Dziubański, J., Preisner, M.: Multiplier theorem for Hankel transform on Hardy spaces. Monatshefte Math. 159, 1–12 (2010) MATHCrossRefGoogle Scholar
  10. 10.
    Gosselin, J., Stempak, K.: A weak-type estimate for Fourier-Bessel multipliers. Proc. Am. Math. Soc. 106(3), 655–662 (1989) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Garrigós, G., Seeger, A.: Characterizations of Hankel multipliers. Math. Ann. 342(1), 31–68 (2008) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Gasper, G., Trebels, W.: Necessary conditions for Hankel multipliers. Indiana Univ. Math. J. 31(3), 403–414 (1982) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Haimo, D.T.: Integral equations associated with Hankel convolutions. Trans. Am. Math. Soc. 116, 330–375 (1965) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Hebisch, W., Zienkiewicz, J.: Multiplier theorems on generalized Heisenberg groups II. Colloq. Math. 69(1), 29–36 (1995) MathSciNetMATHGoogle Scholar
  15. 15.
    Hörmander, L.: Estimates for translation invariant operators in L p spaces. Acta Math. 104, 93–140 (1960) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Lebedev, N.N.: Special Functions and Their Applications. Dover, New York (1972) MATHGoogle Scholar
  17. 17.
    Martini, A.: Algebras of differential operators on Lie groups and spectral multipliers. Ph.D. Thesis, Scuola Normale Superiore Pisa (2009) Google Scholar
  18. 18.
    Mauceri, G., Meda, S.: Vector-valued multipliers on stratified groups. Rev. Mat. Iberoam. 6(3–4), 141–154 (1990) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Müller, D., Stein, E.M.: On spectral multipliers for Heisenberg and related groups. J. Math. Pures Appl. 73, 413–440 (1994) MathSciNetMATHGoogle Scholar
  20. 20.
    Sikora, A.: Multivariable spectral multipliers and analysis of quasielliptic operators on fractals. Indiana Univ. Math. J. 58(1), 317–334 (2009) MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Stein, E.: Topics in Harmonic Analysis Related to the Littlewood-Paley Theory. Princeton University Press, Princeton (1970) MATHGoogle Scholar
  22. 22.
    Stein, E.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970) MATHGoogle Scholar
  23. 23.
    Titchmarsh, E.C.: Introduction to the Theory of Fourier Integrals. Clareoton Press, Oxford (1937) Google Scholar
  24. 24.
    Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam (1978) Google Scholar
  25. 25.
    Uchiyama, A.: A maximal function characterization of H p on the space of homogeneous type. Trans. Am. Math. Soc. 262(2), 579–592 (1980) MathSciNetMATHGoogle Scholar
  26. 26.
    Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge (1944) MATHGoogle Scholar
  27. 27.
    Wróbel, B.: Multivariate spectral multipliers for tensor product orthogonal expansions. Monatshefte Math. 168(1), 125–149 (2012) MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Jacek Dziubański
    • 1
  • Marcin Preisner
    • 1
  • Błażej Wróbel
    • 1
  1. 1.Instytut MatematycznyUniwersytet WrocławskiWrocławPoland

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