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ON AN INTEGRAL OPERATOR FROM THE ZYGMUND SPACE TO THE BLOCH-TYPE SPACE ON THE UNIT BALL

Published online by Cambridge University Press:  01 May 2009

STEVO STEVIĆ*
Affiliation:
Mathematical Institute of the Serbian Academy of Science, Knez Mihailova 36/III, 11000 Beograd, Serbia e-mail: sstevic@ptt.rs
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Abstract

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In this paper, we introduce an integral operator on the unit ball . The boundedness and compactness of the operator from the Zygmund space to the Bloch-type space or the little Bloch-type space are investigated.

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 2008

References

REFERENCES

1.Avetisyan, K., Hardy Bloch-type spaces and lacunary series on the polydisk, Glasgow J. Math. 49 (2007), 345356.CrossRefGoogle Scholar
2.Boe, B. and Nikolau, A., Interpolation by functions in the Bloch space, J. Anal. Math. 94 (2004), 171194.CrossRefGoogle Scholar
3.Chang, D. C., Li, S. and Stević, S., On some integral operators on the unit polydisk and the unit ball, Taiwanese J. Math. 11 (5) (2007), 12511286.Google Scholar
4.Clahane, D. and Stević, S., Norm equivalence and composition operators between Bloch/Lipschitz spaces of the unit ball J. Inequal. Appl. 2006 (2006), 11 pp. (Article ID 61018).Google Scholar
5.Cowen, C. and MacCluer, B., Composition operators on spaces of analytic functions (Studies in Advanced Mathematics, CRC Press, Boca Raton, 1995).Google Scholar
6.Duren, P., Theory of H p spaces (Academic press, New York, 1973).Google Scholar
7.Fu, X. and Zhu, X., Weighted composition operators on some weighted spaces in the unit ball, Abstr. Appl. Anal. 2008 (2008), 8 pp. (Article ID 605807).Google Scholar
8.Hornor, W. and Jamison, J. E., Isometries of some Banach spaces of analytic functions, Integral Equations Operator Theory 41 (2001), 401425.CrossRefGoogle Scholar
9.Hu, Z., Extended Cesàro operators on mixed-norm spaces, Proc. Amer. Math. Soc. 131 (7) (2003), 21712179.CrossRefGoogle Scholar
10.Hu, Z., Extended Cesàro operators on the Bloch space in the unit ball of ℂn, Acta Math. Sci. Ser. B Engl. Ed. 23 (4) (2003), 561566.Google Scholar
11.Hu, Z., Extended Cesàro operators on Bergman spaces, J. Math. Anal. Appl. 296 (2004), 435454.CrossRefGoogle Scholar
12.Li, S. and Stević, S., Integral type operators from mixed-norm spaces to α-Bloch spaces, Integral Transform. Spec. Funct. 18 (7) (2007), 485493.Google Scholar
13.Li, S. and Stević, S., Riemann–Stieltjes operators on Hardy spaces in the unit ball of ℂn, Bull. Belg. Math. Soc. Simon Stevin 14 (2007), 621628.Google Scholar
14.Li, S. and Stević, S., Riemann–Stieltjes type integral operators on the unit ball in ℂn, Complex Variables Elliptic Equations 52 (6) (2007), 495517.Google Scholar
15.Li, S. and Stević, S., Weighted composition operators from Bergman-type spaces into Bloch spaces, Proc. Indian Acad. Sci. Math. Sci. 117 (3) (2007), 371385.CrossRefGoogle Scholar
16.Li, S. and Stević, S., Weighted composition operators from α-Bloch space to H on the polydisk, Numer. Funct. Anal. Optimization 28 (7) (2007), 911925.CrossRefGoogle Scholar
17.Li, S. and Stević, S., Weighted composition operators from H to the Bloch space on the polydisc, Abstr. Appl. Anal. 2007 (2007), 12 pp. (Article ID 48478).CrossRefGoogle Scholar
18.Li, S. and Stević, S., Compactness of Reimann–Stieltjes operators between F(p, q, s) and α-Bloch spaces, Publ. Math. Debrecen 72 (1–2) (2008), 111128.CrossRefGoogle Scholar
19.Li, S. and Stević, S., Generalized composition operators on Zygmund spaces and Bloch type spaces, J. Math. Anal. Appl. 338 (2008), 12821295.CrossRefGoogle Scholar
20.Li, S. and Stević, S., Riemann–Stieltjes operators between mixed norm spaces, Indian J. Math. 50 (1) (2008), 177188.Google Scholar
21.Li, S. and Stević, S., Riemann–Stieltjes operators between different weighted Bergman spaces, Bull. Belg. Math. Soc. Simon Stevin 15 (to appear).Google Scholar
22.Li, S. and Stević, S., Weighted composition operators between H and α-Bloch spaces in the unit ball, Taiwanese J. Math. (accepted for publication).Google Scholar
23.Luo, L. and Ueki, S. I., Weighted composition operators between weighted Bergman spaces and Hardy spaces on the unit ball of ℂn, J. Math. Anal. Appl. 326 (1) (2007), 88100.Google Scholar
24.MacCluer, B. D. and Zhao, R., Essential norms of weighted composition operators between Bloch-type spaces, Rocky Mountain J. Math. 33 (4) (2003), 14371458.CrossRefGoogle Scholar
25.Madigan, K. and Matheson, A., Compact composition operators on the Bloch space, Trans. Amer. Math. Soc. 347 (7) (1995), 26792687.Google Scholar
26.Ohno, S., Weighted composition operators between H and the Bloch space, Taiwanese J. Math. 5 (2001), 555563.CrossRefGoogle Scholar
27.Ohno, S., Stroethoff, K. and Zhao, R., Weighted composition operators between Bloch type spaces, Rocky Mountain J. Math. 33 (2003), 191215.CrossRefGoogle Scholar
28.Rudin, W., Function theory in the unit ball of ℂn (Springer-Verlag, New York, 1980).Google Scholar
29.Shi, J. H. and Luo, L., Composition operators on the Bloch space, Acta Math. Sinica 16 (2000), 8598.Google Scholar
30.Shields, A. L. and Williams, D. L., Bounded projections, duality, and multipliers in spaces of analytic functions, Trans. Amer. Math. Soc. 162 (1971), 287302.Google Scholar
31.Stević, S., On an integral operator on the unit ball in ℂn, J. Inequal. Appl. 1 (2005), 8188.Google Scholar
32.Stević, S., Boundedness and compactness of an integral operator on a weighted space on the polydisc, Indian J. Pure Appl. Math. 37 (6) (2006), 343355.Google Scholar
33.Stević, S., Composition operators between H and the α-Bloch spaces on the polydisc, Z. Anal. Anwendungen 25 (4) (2006), 457466.CrossRefGoogle Scholar
34.Stević, S., Boundedness and compactness of an integral operator on mixed norm spaces on the polydisc, Sibirsk. Mat. Zh. 48 (3) (2007), 694706.Google Scholar
35.Stević, S., On α-Bloch spaces with Hadamard gaps, Abstr. Appl. Anal. 2007 (2007), 7 pp. (Article ID 39176).CrossRefGoogle Scholar
36.Stević, S., Weighted composition operators between mixed norm spaces and H α spaces in the unit ball, J. Inequal. Appl. 2007 (2007), 9 pp. (Article ID 28629).Google Scholar
37.Stević, S., On a new integral-type operator from the weighted Bergman space to the Bloch-type space on the unit ball, Discrete Dyn. Nat. Soc. 2008 (2008), 14 pp. (Article ID 154263).Google Scholar
38.Stević, S., Generalized composition operators from logarithmic Bloch spaces to mixed-norm spaces, Util. Math. 77 (to appear)Google Scholar
39.Stević, S., Extended Cesàro operators between mixed-norm spaces and Bloch-type spaces in the unit ball, Houston J. Math. (to appear).Google Scholar
40.Tang, X., Extended Cesàro operators between Bloch-type spaces in the unit ball of ℂn, J. Math. Anal. Appl. 326 (2) (2007), 11991211.Google Scholar
41.Ueki, S. I. and Luo, L., Compact weighted composition operators and multiplication operators between Hardy spaces, Abstr. Appl. Anal. 2008 (2008), 11 pp. (Article ID 196498).Google Scholar
42.Xiao, J., Riemann–Stieltjes operators on weighted Bloch and Bergman spaces of the unit ball, J. Lond. Math. Soc. 70 (2) (2004), 199214.CrossRefGoogle Scholar
43.Yamashita, S., Gap series and α-Bloch functions, Yokohama Math. J. 28 (1980), 3136.Google Scholar
44.Ye, S., Weighted composition operators between the little α-Bloch space and the logarithmic Bloch, J. Comput. Anal. Appl. 10 (2) (2008), 243252.Google Scholar
45.Zhu, K., Spaces of holomorphic functions in the unit ball (Graduate Text in Mathematics 226, Springer, New York, 2005).Google Scholar