Journal d'Analyse Mathématique

, Volume 128, Issue 1, pp 261–287 | Cite as

Hardy spaces for Fourier-Bessel expansions

  • Jacek DziubańskiEmail author
  • Marcin Preisner
  • Luz Roncal
  • Pablo Raúl Stinga


We study Hardy spaces for Fourier-Bessel expansions associated with Bessel operators on \(((0,1),{x^{2\nu + 1}}dx)\) and ((0, 1), dx). We define Hardy spaces H 1 as the sets of L 1-functions whose maximal functions for the corresponding Poisson semigroups belong to L 1. Atomic characterizations are obtained.


Hardy Space Maximal Function Homogeneous Type Atomic Decomposition Poisson Kernel 
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  1. [1]
    A. Benedek and R. Panzone, On mean convergence of Fourier-Bessel series of negative order, Studies in Appl. Math. 50 (1971), 281–292.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    A. Benedek and R. Panzone, On convergence of orthogonal series of Bessel functions, Ann. Scuola Norm. Sup. Pisa (3) 27 (1973), 505–525.MathSciNetzbMATHGoogle Scholar
  3. [3]
    J. J. Betancor, J. Dziubański, and J. L. Torrea, On Hardy spaces associated with Bessel operators, J. Anal. Math. 107 (2009), 195–219.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Ó. Ciaurri and L. Roncal, The Bochner-Riesz means for Fourier-Bessel expansions, J. Funct. Anal. 228 (2005), 89–113.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Ó. Ciaurri and L. Roncal, Littlewood-Paley-Stein gk-functions for Fourier-Bessel expansions, J. Funct. Anal. 258 (2010), 2173–2204.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Ó. Ciaurri and L. Roncal, Higher order Riesz transforms for Fourier-Bessel expansions, J. Fourier Anal. Appl. 18 (2012), 770–789.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Ó. Ciaurri and K. Stempak, Conjugacy for Fourier-Bessel expansions, Studia Math. 176 (2006), 215–247.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Ó. Ciaurri and K. Stempak, Transplantation and multiplier theorems for Fourier-Bessel expansions, Trans. Amer.Math. Soc. 358 (2006), 4441–4465.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Ó. Ciaurri and K. Stempak, Weighted transplantation for Fourier-Bessel expansions, J. Anal. Math. 100 (2006), 133–156.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    R. Coifman, A real variable characterization of Hp, Studia Math. 51 (1974), 269–274.MathSciNetzbMATHGoogle Scholar
  11. [11]
    R. Coifman and G. Weiss, Analyse Harmonique Non-Commutative sur Certains Espaces Homogènes, Springer, Berlin, 1971.zbMATHGoogle Scholar
  12. [12]
    J. Dziubański, M. Preisner, and B. Wróbel, Multivariate Hörmander-type multiplier theorem for the Hankel transform, J. Fourier Anal. Appl. 19 (2013), 417–437.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    J. Dziubański and J. Zienkiewicz, Hardy space H1 associated to Schrödinger operator with potential satisfying reverse Hölder inequality, Rev. Mat. Iberoamericana 15 (1999), 279–296.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    C. Fefferman and E. M. Stein, Hp spaces of several variables, Acta Math. 129 (1972), 137–193.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    G. B. Folland, Introduction to Partial Differential Equations, 2nd ed., Princeton University Press, Princeton, NJ, 1995.zbMATHGoogle Scholar
  16. [16]
    J. E. Gilbert, Maximal theorems for some orthogonal series I, Trans. Amer. Math. Soc. 145 (1969), 495–515.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    D. Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 (1979), 27–42.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    R. Latter, A decomposition of Hp(Rn) in terms of atoms, Studia Math. 62 (1978), 93–101.MathSciNetzbMATHGoogle Scholar
  19. [19]
    N. N. Lebedev, Special Functions and Their Applications, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1965.zbMATHGoogle Scholar
  20. [20]
    R. A. Macías and C. Segovia, A decomposition into atoms of distributions on spaces of homogeneous type, Adv. Math. 33 (1979), 271–309.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    A. Nowak and L. Roncal, On sharp heat and subordinated kernel estimates in the Fourier-Bessel setting, Rocky Mountain J. Math. 44 (2014), 1321–1342.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    A. Nowak and L. Roncal, Sharp heat kernel estimates in the Fourier-Bessel setting for a continuous range of the type parameter, Acta Math. Sinica, 30 (2014), 437–444.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    A. Nowak and L. Roncal, Potential operators associated with Jacobi and Fourier-Bessel expansions, J. Math. Anal. Appl. 422 (2015), 148–184.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    M. Preisner, Riesz transform characterization of H 1 spaces associated with certain Laguerre expansions, J. Approx. Theory 164 (2012), 229–252.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993.zbMATHGoogle Scholar
  26. [26]
    E. M. Stein and G. Weiss, On the theory of harmonic functions of several variables, I: the theory of Hp spaces, Acta Math. 103 (1960), 25–62.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    A. Uchiyama, A maximal function characterization of Hp on the space of homogeneous type, Trans. Amer. Math. Soc. 262 (1980), 579–592.MathSciNetzbMATHGoogle Scholar
  28. [28]
    G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1995.zbMATHGoogle Scholar

Copyright information

© Hebrew University Magnes Press 2016

Authors and Affiliations

  • Jacek Dziubański
    • 1
    Email author
  • Marcin Preisner
    • 1
  • Luz Roncal
    • 2
  • Pablo Raúl Stinga
    • 3
  1. 1.InstytutmatematycznyUniwersytetwrocławskiWrocław, Pl. Grunwaldzki 2/4Poland
  2. 2.Departamento de Matemáticas Y ComputaciónUniversidad de la RiojaLogroñoSpain
  3. 3.Department of MathematicsIowa State UniversityAmesUSA

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