On the generalized Hardy spaces. (English) Zbl 1191.30021

Let \(H(U)\) be the space of all analytic functions on the unit disk \(U\) of the complex plane. Given \(1\leq p<\infty\) and a one-to-one linear operator \(F:H(U)\to H(U)\), the author introduces the space \(H_{F, p}(U)\) consisting of all functions \(f\in H(U)\) such that
\[ \|f\|^p_{H_{F,p}}:=\sup_{0<r<1}\int_0^{2\pi}|F(f)(re^{i\theta})|^p\, {{d\theta}\over{2\pi}}<\infty. \]
Clearly, if \(F\) is the identity operator, then \(H_{F, p}(U)\) is the classical Hardy space. The author obtains some sufficient conditions that imply some basic properties of such a space, namely, the property of being a Banach space and the continuity of point evaluations. The author also introduces the notion of Carleson measures for \(H_{F,p}(U)\) and uses them to obtain results concerning boundedness and compactness of composition operators on \(H_{F,p}(U)\).


30H10 Hardy spaces
46E15 Banach spaces of continuous, differentiable or analytic functions
47B33 Linear composition operators
Full Text: DOI EuDML


[1] P. L. Duren, Theory of Hp Spaces, vol. 38 of Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1970. · Zbl 0215.20203
[2] C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, Fla, USA, 1995. · Zbl 0873.47017
[3] J. V. Ryff, “Subordinate Hp functions,” Duke Mathematical Journal, vol. 33, pp. 347-354, 1966. · Zbl 0148.30205 · doi:10.1215/S0012-7094-66-03340-0
[4] H. J. Schwartz, Composition operators on Hp, Ph.D. thesis, University of Toledo, 1969. · Zbl 0193.31104
[5] J. H. Shapiro and P. D. Taylor, “Compact, nuclear, and Hilbert-Schmidt composition operators on H2,” Indiana University Mathematics Journal, vol. 23, pp. 471-496, 1973. · Zbl 0276.47037 · doi:10.1512/iumj.1973.23.23041
[6] J. H. Shapiro, Composition Operators and Classical Function Theory, Universitext: Tracts in Mathematics, Springer, New York, NY, USA, 1993. · Zbl 0791.30033
[7] B. R. Choe, H. Koo, and W. Smith, “Composition operators acting on holomorphic Sobolev spaces,” Transactions of the American Mathematical Society, vol. 355, no. 7, pp. 2829-2855, 2003. · Zbl 1031.47017 · doi:10.1090/S0002-9947-03-03273-2
[8] B. R. Choe, H. Koo, and W. Smith, “Composition operators on small spaces,” Integral Equations and Operator Theory, vol. 56, no. 3, pp. 357-380, 2006. · Zbl 1114.47028 · doi:10.1007/s00020-006-1420-x
[9] D. D. Clahane and S. Stević, “Norm equivalence and composition operators between Bloch/Lipschitz spaces of the ball,” Journal of Inequalities and Applications, vol. 2006, Article ID 61018, 11 pages, 2006. · Zbl 1131.47018 · doi:10.1155/JIA/2006/61018
[10] C. C. Cowen and B. D. MacCluer, “Linear fractional maps of the ball and their composition operators,” Acta Scientiarum Mathematicarum, vol. 66, no. 1-2, pp. 351-376, 2000. · Zbl 0970.47011
[11] X. Fu and X. Zhu, “Weighted composition operators on some weighted spaces in the unit ball,” Abstract and Applied Analysis, vol. 2008, Article ID 605807, 8 pages, 2008. · Zbl 1160.47024 · doi:10.1155/2008/605807
[12] S. Li and S. Stević, “Weighted composition operators from Bergman-type spaces into Bloch spaces,” Proceedings of the Indian Academy of Sciences. Mathematical Sciences, vol. 117, no. 3, pp. 371-385, 2007. · Zbl 1130.47016 · doi:10.1007/s12044-007-0032-y
[13] S. Li and S. Stević, “Weighted composition operators from \alpha -Bloch space to H\infty on the polydisc,” Numerical Functional Analysis and Optimization, vol. 28, no. 7-8, pp. 911-925, 2007. · Zbl 1130.47015 · doi:10.1080/01630560701493222
[14] S. Li and S. Stević, “Weighted composition operators from H\infty to the Bloch space on the polydisc,” Abstract and Applied Analysis, vol. 2007, Article ID 48478, 13 pages, 2007. · Zbl 1152.47016 · doi:10.1155/2007/48478
[15] S. Li and S. Stević, “Weighted composition operators from Zygmund spaces into Bloch spaces,” Applied Mathematics and Computation, vol. 206, no. 2, pp. 825-831, 2008. · Zbl 1215.47022 · doi:10.1016/j.amc.2008.10.006
[16] B. D. MacCluer and J. H. Shapiro, “Angular derivatives and compact composition operators on the Hardy and Bergman spaces,” Canadian Journal of Mathematics, vol. 38, no. 4, pp. 878-906, 1986. · Zbl 0608.30050 · doi:10.4153/CJM-1986-043-4
[17] S. Stević, “On generalized weighted Bergman spaces,” Complex Variables. Theory and Application, vol. 49, no. 2, pp. 109-124, 2004. · Zbl 1053.47020 · doi:10.1080/02781070310001650047
[18] S. Stević, “Composition operators between H\infty and \alpha -Bloch spaces on the polydisc,” Zeitschrift für Analysis und ihre Anwendungen, vol. 25, no. 4, pp. 457-466, 2006. · Zbl 1118.47015 · doi:10.4171/ZAA/1301
[19] S. Stević, “Weighted composition operators between mixed norm spaces and H\alpha \infty spaces in the unit ball,” Journal of Inequalities and Applications, vol. 2007, Article ID 28629, 9 pages, 2007. · Zbl 1138.47019 · doi:10.1155/2007/28629
[20] S. Stević, “Essential norms of weighted composition operators from the \alpha -Bloch space to a weighted-type space on the unit ball,” Abstract and Applied Analysis, vol. 2008, Article ID 279691, 11 pages, 2008. · Zbl 1160.32011 · doi:10.1155/2008/279691
[21] S. Stević, “Norm of weighted composition operators from Bloch space to H\mu \infty on the unit ball,” Ars Combinatoria, vol. 88, pp. 125-127, 2008. · Zbl 1224.30195
[22] S. Stević, “Composition operators from the Hardy space to the Zygmund-type space on the upper half-plane,” Abstract and Applied Analysis, vol. 2009, Article ID 161528, 8 pages, 2009. · Zbl 1173.30036 · doi:10.1155/2009/161528
[23] S. Ueki, “Composition operators on the Privalov spaces of the unit ball of \Bbb Cn,” Journal of the Korean Mathematical Society, vol. 42, no. 1, pp. 111-127, 2005. · Zbl 1062.32004 · doi:10.4134/JKMS.2005.42.1.111
[24] S. Ueki, “Weighted composition operators on the Bargman-Fock space,” International Journal of Modern Mathematics, vol. 3, no. 3, pp. 231-243, 2008. · Zbl 1171.47021
[25] S. Ueki and L. Luo, “Compact weighted composition operators and multiplication operators between Hardy spaces,” Abstract and Applied Analysis, vol. 2008, Article ID 196498, 12 pages, 2008. · Zbl 1167.47020 · doi:10.1155/2008/196498
[26] S. Ye, “Weighted composition operator between the little \alpha -Bloch spaces and the logarithmic Bloch,” Journal of Computational Analysis and Applications, vol. 10, no. 2, pp. 243-252, 2008. · Zbl 1152.47019
[27] K. Zhu, “Compact composition operators on Bergman spaces of the unit ball,” Houston Journal of Mathematics, vol. 33, no. 1, pp. 273-283, 2007. · Zbl 1114.47031
[28] X. Zhu, “Weighted composition operators from logarithmic Bloch spaces to a class of weighted-type spaces in the unit ball,” Ars Combinatoria, vol. 91, pp. 87-95, 2009. · Zbl 1216.47043
[29] K. Zhu, Operator Theory in Function Spaces, vol. 139 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1990.
[30] B. D. MacCluer, “Compact composition operators on Hp(BN),” The Michigan Mathematical Journal, vol. 32, no. 2, pp. 237-248, 1985. · Zbl 0585.47022 · doi:10.1307/mmj/1029003191
[31] W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, NY, USA, 3rd edition, 1987. · Zbl 0925.00005
[32] R. G. Bartle, The Elements of Real Analysis, John Wiley & Sons, New York, NY, USA, 2nd edition, 1976. · Zbl 0309.26003
[33] P. R. Halmos, Measure Theory, Springer, New York, NY, USA, 1974. · Zbl 0283.28001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.