THE ATOMIC DECOMPOSITION OF HARMONIC BERGMAN FUNCTIONS, DUALITIES AND TOEPLITZ OPERATORS

Title & Authors
THE ATOMIC DECOMPOSITION OF HARMONIC BERGMAN FUNCTIONS, DUALITIES AND TOEPLITZ OPERATORS
Lee, Young-Joo;

Abstract
On the setting of the unit ball of $\small{{\mathbb{R}}^n}$, 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
atomic decomposition;harmonic Bergman space;Toeplitz operator;
Language
English
Cited by
1.
The Bloch space and the dual space of a Luecking-type subspace of , Complex Variables and Elliptic Equations, 2017, 1
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

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

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

11.
J. Miao, Reproducing kernels for harmonic Bergman spaces of the unit ball, Monatsh. Math. 125 (1998), no. 1, 25–35

12.
J. Miao, Toeplitz operators on harmonic Bergman spaces, Integral Equations Operator Theory 27 (1997), no. 4, 426–438

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