Toeplitz operators on harmonic Bergman spaces. (English) Zbl 0902.47026

Summary: We study Toeplitz operators on the harmonic Bergman space \(b^p(B)\), where \(B\) is the open unit ball in \(\mathbb{R}^n\) \((n\geq 2)\), for \(1< p<\infty\). We give characterizations for the Toeplitz operators with positive symbols to be bounded, compact, and in Schatten classes. We also obtain a compactness criteria for the Toeplitz operators with continuous symbols.


47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
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[1] J. Arazy, S. Fisher, and J. Peetre,Hankel operators on weighted Bergman spaces, Amer. J. Math. 110 (1988), 989-1054. · Zbl 0669.47017
[2] S. Axler,Bergman spaces and their operators, Surveys of Some Results in Operator Theory, vol. 1, (J.B. Conway and B.B. Morrel, editors), Pitman Res. Notes Math. Ser. vol. 171, 1988, pp. 1-50.
[3] S. Axler, P. Bourdon, and W. Ramey,Harmonic Function Theory, Springer-Verlag, New York, 1992. · Zbl 0765.31001
[4] R.R. Coifman, R. Rochberg,Representation theorems for holomorphic and harmonic functions, Ast?risque 77 (1980), 11-65. · Zbl 0472.46040
[5] M. Jovovi?,Compact Hankel operators on harmonic Bergman spaces, Integral Equations and Operator Theory 22 (1995), 295-304. · Zbl 0837.47021
[6] E. Ligocka,On the reproducing kernel for harmonic functions and the space of Bloch harmonic functions on the unit ball in R n , Studia Math. 87 (1987), 23-32. · Zbl 0658.31006
[7] E. Ligocka, Estimates in Sobolev norms ??? p 3 for harmonic and holomorphic fuctions and interpolation between Sobolev and Holder spaces of harmonic functions, Studia Math. 86 (1987), 255-271. · Zbl 0642.46035
[8] D.H. Luecking,Trace ideal criteria for Toeplity operators, J. Funct. Anal. 73 (1987), 345-368. · Zbl 0618.47018
[9] D.H. Luecking,Characterizations of certain classes of Hankel operators on the Bergman spaces of the unit disc, J. Funct. Anal. 110 (1992), 247-271. · Zbl 0773.47014
[10] O.L. Oleinik,Embedding theorems for weighted classes of harmonic and analytic functions, J. Soviet Math. 9 (1978), 228-243. · Zbl 0396.31001
[11] V.L. Oleinik, B.S. Pavlov,Embedding theorems for weighted classes of harmonic and analytic functions, J. Soviet Math. 2 (1974), 135-142. · Zbl 0278.46032
[12] E.M. Stein,Singular intergrals and differentiability properties of functions, Princeton University Press, Princeton, NJ, 1970. · Zbl 0207.13501
[13] Z. Wu,Operators on harmonic Bergman spaces, Preprint. · Zbl 0860.47017
[14] K. Zhu,Positive Toeplitz operators on weighted Bergman spaces of bounded symmetric domains, J. Operator Theory 20 (1988), 329-357. · Zbl 0676.47016
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