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, Volume 71, Issue 1, pp 67–81 | Cite as

M-harmonic Besovp-spaces and Hankel operators in the Bergman space on the ball in ℂ n

  • Kyong T. Hahn
  • E. H. Youssfi


In this paper, we characterize the Hardy class ofM-harmonic functions on the unit ballB in ℂ n in terms of the Berezin transform. We define and study the Besovp-spaces ofM-harmonic functions. For anM-harmonic symbolf, we give various criteria for the Hankel operatorsH f andH f to be bounded, compact or in the Schatten-von-Neumann classS p . These criteria establish a close relationship among Besovp-spaces, Berezin transform, the invariant Laplacian, and Hankel operators on the unit ballB.


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  1. [1]
    J. Arazy, S. Fisher andJ. Peetre,Möbius invariant function spaces, J. für die reine und angewandte Math., 363 (1985), pp. 110–145MathSciNetMATHGoogle Scholar
  2. [2]
    J. Arazy, S. Fisher andJ. Peetre,Hankel operators on weighted Bergman spaces, Amer. J. Math., 110 (1988), pp. 989–1054MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    S. Axler,The Bergman space, the Bloch space and commutators of multiplication operators, Duke Math. J., 53 (1986), pp. 315–332MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    C. A. Berger, L. A. Coburn andK. H. Zhu,Function theory in Cartan domains and the Berezin-Toeplitz symbol calculus, Amer. J. Math., 110 (1988), pp. 921–953MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    C. A. Berger, L. A. Coburn, andK. H. Zhu,BMO on the Bergman spaces of the classical domains, Bull. Amer. Math. Soc., 17, 1 (1987), pp. 133–136MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    K. T. Hahn,Holomorphic mappings of the hyperbolic space into the complex Euclidien space and the Bloch theorem, Can. J. Math., 27 (1975) pp. 446–458CrossRefMATHGoogle Scholar
  7. [7]
    K. T. Hahn and E. H. Youssfi,Möbius invariant Besov p-spaces and Hankel Operators in the Bergman space on the ball in ℂ n, compl. Var. Th. Appl. (To appear)Google Scholar
  8. [8]
    A. Kóranyi,Harmonic functions on hermitian hyperbolic space, Trans. Amer. Math. Soc.,135 (1969), pp. 507–516MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    W. Rudin,Function theory in the unit ball of ℂ n, Springer-Verlag, 1980Google Scholar
  10. [10]
    R. M. Timoney,Bloch functions in several complex variables I, Bull. London Math. Soc., 12 (1980), pp. 241–267MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    D. Ullrich,Moebius invariant potential theory in the unit ball of ℂ n,Thesis, Univ. of Wisconson Madison 1981, University Microfilms International, Ann Arbor, Michigan, 1984Google Scholar
  12. [12]
    D. Zheng,Schatten class Hankel operators on the Bergman space, Integral Equations and Operator Theory, 13 (1990), pp. 442–459MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    K. H. Zhu,Hilbert-Schmidt Hankel operators on the Bergman space, Proc. Amer. Math. Soc., 109 (1990), pp. 721–730MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    K.H. Zhu,Schatten class Hankel operators on the Bergman space of the unit ball, preprint Google Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Kyong T. Hahn
    • 1
  • E. H. Youssfi
    • 1
  1. 1.Department of MathematicsThe Pennsylvania State UniversityUniversity Park

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