Journal d'Analyse Mathématique

, Volume 107, Issue 1, pp 195–219 | Cite as

On Hardy spaces associated with Bessel operators

  • Jorge J. BetancorEmail author
  • Jacek Dziubański
  • Jose Luis Torrea


In this paper, we study Hardy spaces associated with two Bessel operators. Two different kind of Hardy spaces appear. These differences are transparent in the corresponding atomic decompositions.


Hardy Space Maximal Function Hardy Inequality Homogeneous Type Atomic Decomposition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    K. F. Andersen and B. Muckenhoupt, Weighted weak type Hardy inequalities with applications to Hilbert transforms and maximal functions, Studia Math. 72 (1982), 9–26.zbMATHMathSciNetGoogle Scholar
  2. [2]
    J. J. Betancor, D. Buraczewski, J. C. Fariña, M. T. Martínez and J. L. Torrea, Riesz transforms related to Bessel operators, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), 701–725.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    J. J. Betancor, J. C. Fariña and A. Sanabria, On Littlewood-Paley functions associated with Bessel operators, Glasgow Math. J., to appear.Google Scholar
  4. [4]
    R. Coifman and G. Weiss, Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes, Lecture Notes in Math. 242, Springer-Verlag, Berlin-New York, 1971.Google Scholar
  5. [5]
    R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569–615.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    J. Dziubański, Hardy spaces associated with semigroups generated by Bessel operators with potentials, Houston J. Math. 34 (2008), 205–234.zbMATHMathSciNetGoogle Scholar
  7. [7]
    S. Fridli, Hardy spaces generated by an integrability condition, J. Approx. Theory 113 (2001), 91–109.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    J. García-Cuerva, Weighted H p spaces, Dissertationes Math. (Rozprawy Mat.) 162 (1979).Google Scholar
  9. [9]
    D. T. Haimo, Equations associated with Hankel convolutions, Trans. Amer. Math. Soc. 116 (1965), 330–375.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    I. I. Hirschman, Variation diminishing Hankel transforms, J. Analyse Math. 8 (1960/1961), 307–336.CrossRefMathSciNetGoogle Scholar
  11. [11]
    L. Hörmander, Estimates for translation invariant operators in L p spaces, Acta. Math. 104 (1960), 93–140.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    Y. Kanjin, On Hardy inequalities and Hankel transforms, Monatsh. Math. 127 (1999), 311–319.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    Y. Kanjin, A transplantation theorem for the Hankel transform on the Hardy space, Tohoku Math. J. (2) 57 (2005), 231–246.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    N. N. Lebedev, Special Functions and Their Applications, Dover, New York, 1972.zbMATHGoogle Scholar
  15. [15]
    P. Macaulay-Owen, Parseval’s theorem for Hankel transforms, Proc. London Math. Soc. 45 (1939), 458–474.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    R. Macías and C. Segovia, A decomposition into atoms of distributions on spaces of homogeneous type, Adv. in Math. 33 (1979), 271–309.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    B. Muckenhoupt and E. Stein, Classical expansions and their relation to conjugate harmonic functions, Trans. Amer. Math. Soc. 118 (1965), 17–92.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    S. Schindler, Explicit integral transform proofs of some transplantation theorems for the Hankel transform, SIAM J. Math. Anal. 4 (1973), 367–384.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    E. M. Stein, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory, Princeton University Press, Princeton, NJ, 1970.zbMATHGoogle Scholar
  20. [20]
    E. M. Stein, Harmonic Analysis, Real Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993.zbMATHGoogle Scholar
  21. [21]
    K. Stempak, The Littlewood-Paley theory for the Fourier-Bessel transform, preprint no 45, Math. Inst. Univ. Wroclaw, Poland, 1985.Google Scholar
  22. [22]
    A. Uchiyama, A maximal function characterization of H p on the space of homogeneous type, Trans. Amer. Math. Soc. 262 (1980), 579–592.zbMATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1966.zbMATHGoogle Scholar
  24. [24]
    A. Weinstein, Discontinuous integrals and generalized potential theory, Trans. Amer. Math. Soc. 63 (1948), 342–354.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Hebrew University Magnes Press 2009

Authors and Affiliations

  • Jorge J. Betancor
    • 1
    Email author
  • Jacek Dziubański
    • 2
  • Jose Luis Torrea
    • 3
  1. 1.Departamento De Análisis MatemáticoUniversidad De La LagunaLa Laguna (Sta. Cruz de Tenerife)Spain
  2. 2.Institute Of MathematicsUniversity Of WroclawWroclawPoland
  3. 3.Departamento De MatemáticasFacultad De Ciencias Universidad Autónoma De MadridMadridSpain

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