Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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THE ATOMIC DECOMPOSITION OF HARMONIC BERGMAN FUNCTIONS, DUALITIES AND TOEPLITZ OPERATORS

  • Lee, Young-Joo (DEPARTMENT OF MATHEMATICS CHONNAM NATIONAL UNIVERSITY)
  • Published : 2009.03.31
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Abstract

On the setting of the unit ball of ${\mathbb{R}}^n$, we consider a Banach space of harmonic functions motivated by the atomic decomposition in the sense of Coifman and Rochberg [5]. First we identify its dual (resp. predual) space with certain harmonic function space of (resp. vanishing) logarithmic growth. Then we describe these spaces in terms of boundedness and compactness of certain Toeplitz operators.

Keywords

References

  1. S. Axler, P. Bourdon, and W. Ramey, Harmonic Function Theory, Springer-Verlag, New York, 1992
  2. B. R. Choe, Note on the Berezin transform on Herz spaces, RIMS Kyokuroku, 21-37, 2006
  3. B. R. Choe and Y. J. Lee, A Luecking type subspace, dualities and Toeplitz operators, Acta Math. Hungar. 67 (1995), no. 1-2, 151–170 https://doi.org/10.1007/BF01874529
  4. B. R. Choe, Y. J. Lee, and K. Na, Toeplitz operators on harmonic Bergman spaces, Nagoya Math. J. 174 (2004), 165–186
  5. R. R. Coifman and R. Rochberg, Representation theorems for holomorphic and harmonic functions in $L^p$, Representation theorems for Hardy spaces, pp. 11–66, Asterisque, 77, Soc. Math. France, Paris, 1980
  6. P. Duren and A. Schuster, Bergman Space, American Mathematical Society, 2004
  7. P. Ghatage and S. Sun, A Luecking-type subspace of $L^1_a$ and its dual, Proc. Amer. Math. Soc. 110 (1990), no. 3, 767–774
  8. P. Ghatage and S. Sun, Duality and multiplication operators, Integral Equations Operator Theory 14 (1991), no. 2, 213–228 https://doi.org/10.1007/BF01199906
  9. W. K. Hayman and P. B. Kennedy, Subharmonic Functions. Vol. I, London Mathematical Society Monographs, No. 9. Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976
  10. M. Jevtic and M. Pavlovic, Harmonic Bergman functions on the unit ball in $R^n$, Acta Math. Hungar. 85 (1999), no. 1-2, 81–96 https://doi.org/10.1023/A:1006620929091
  11. J. Miao, Reproducing kernels for harmonic Bergman spaces of the unit ball, Monatsh. Math. 125 (1998), no. 1, 25–35 https://doi.org/10.1007/BF01489456
  12. J. Miao, Toeplitz operators on harmonic Bergman spaces, Integral Equations Operator Theory 27 (1997), no. 4, 426–438 https://doi.org/10.1007/BF01192123
  13. M. Pavlovic, Decompositions of $L^p$ and Hardy spaces of polyharmonic functions, J. Math. Anal. Appl. 216 (1997), no. 2, 499–509
  14. S. Perez-Esteva, Duality on vector-valued weighted harmonic Bergman spaces, Studia Math. 118 (1996), no. 1, 37–47

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