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\(\mathcal M\)-harmonic Besov \(p\)-spaces and Hankel operators in the Bergman space on the ball in \(\mathbb{C}^ n\). (English) Zbl 0816.31004
The authors establish a reproducing formula for \({\mathcal M}\)-harmonic functions in \(h^ p (B)\), the space of all \({\mathcal M}\)-harmonic functions \(f\) on the unit ball \(B = B^ n \subseteq \mathbb{C}^ n\) that satisfy a growth condition. Extending the definition of Besov \(p\)-spaces to \({\mathcal M}\)-harmonic functions on \(B\), they obtain several characterizations of \({\mathcal M} B_ p (B)\). Various equivalent conditions are also obtained for the Hankel operators \(H_ f\) and \(H_{\overline f}\) to be bounded, compact or to lie in the Schatten-von-Neumann class.
Some corrections are given in the Erratum (Zbl 0816.31005) stated below.
Reviewer: P.C.Sinha (Patna)

31C05 Harmonic, subharmonic, superharmonic functions on other spaces
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
Full Text: DOI EuDML
[1] J. Arazy, S. Fisher andJ. Peetre,Möbius invariant function spaces, J. für die reine und angewandte Math., 363 (1985), pp. 110–145 · Zbl 0566.30042
[2] J. Arazy, S. Fisher andJ. Peetre,Hankel operators on weighted Bergman spaces, Amer. J. Math., 110 (1988), pp. 989–1054 · Zbl 0669.47017 · doi:10.2307/2374685
[3] S. Axler,The Bergman space, the Bloch space and commutators of multiplication operators, Duke Math. J., 53 (1986), pp. 315–332 · Zbl 0633.47014 · doi:10.1215/S0012-7094-86-05320-2
[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. 921–953 · Zbl 0657.32001 · doi:10.2307/2374698
[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. 133–136 · Zbl 0621.32014 · doi:10.1090/S0273-0979-1987-15539-X
[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. 446–458 · Zbl 0299.32017 · doi:10.4153/CJM-1975-053-0
[7] K. T. Hahn and E. H. Youssfi,Möbius invariant Besov p-spaces and Hankel Operators in the Bergman space on the ball in \(\mathbb{C}\) n , compl. Var. Th. Appl. (To appear) · Zbl 0706.47017
[8] A. Kóranyi,Harmonic functions on hermitian hyperbolic space, Trans. Amer. Math. Soc.,135 (1969), pp. 507–516 · Zbl 0174.38801
[9] W. Rudin,Function theory in the unit ball of \(\mathbb{C}\) n , Springer-Verlag, 1980 · Zbl 0495.32001
[10] R. M. Timoney,Bloch functions in several complex variables I, Bull. London Math. Soc., 12 (1980), pp. 241–267 · Zbl 0428.32018 · doi:10.1112/blms/12.4.241
[11] D. Ullrich,Moebius invariant potential theory in the unit ball of \(\mathbb{C}\) n ,Thesis, Univ. of Wisconson Madison 1981, University Microfilms International, Ann Arbor, Michigan, 1984
[12] D. Zheng,Schatten class Hankel operators on the Bergman space, Integral Equations and Operator Theory, 13 (1990), pp. 442–459 · Zbl 0729.47023 · doi:10.1007/BF01199895
[13] K. H. Zhu,Hilbert-Schmidt Hankel operators on the Bergman space, Proc. Amer. Math. Soc., 109 (1990), pp. 721–730 · Zbl 0731.47028 · doi:10.1090/S0002-9939-1990-1013987-7
[14] K.H. Zhu,Schatten class Hankel operators on the Bergman space of the unit ball, preprint · Zbl 0734.47017
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