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Stević, Stevo; Ueki, Sei-Ichiro

On an integral-type operator acting between Bloch-type spaces on the unit ball. (English) Zbl 1200.32005

Abstr. Appl. Anal. 2010, Article ID 214762, 14 p. (2010).
Let \(\mathbb B\) denote the open unit ball of \(\mathbb C^n\). For a holomorphic self-map \(\varphi\) of \(\mathbb B\) and a holomorphic function \(g\) in \(\mathbb B\) with \(g(0)=0\), we define the following integral-type operator: \(I_\varphi^g f(z)=\int^1_0 \operatorname{Re} f(\varphi(tz))g(tz)(dt/t)\), \(z\in\mathbb B\). Here \(\operatorname{Re} f\) denotes the radial derivative of a holomorphic function \(f\) in \(\mathbb B\). We study the boundedness and compactness of the operator between Bloch-type spaces \({\mathcal B}\omega\) and \({\mathcal B}\mu\), where \(\omega\) is a normal weight function and \(\mu\) is a weight function. Also, we consider the operator between the little Bloch-type spaces \({\mathcal B}_{\omega,0}\) and \({\mathcal B}_{\mu,0}\).

Cited in 10 Documents

MSC:

32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))
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\textit{S. Stević} and \textit{S.-I. Ueki}, Abstr. Appl. Anal. 2010, Article ID 214762, 14 p. (2010; Zbl 1200.32005)
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References:

[1] S. Stević, “On an integral operator from the Zygmund space to the Bloch-type space on the unit ball,” Glasgow Mathematical Journal, vol. 51, no. 2, pp. 275-287, 2009. · Zbl 1176.47029
[2] S. Li and S. Stević, “Generalized composition operators on Zygmund spaces and Bloch type spaces,” Journal of Mathematical Analysis and Applications, vol. 338, no. 2, pp. 1282-1295, 2008. · Zbl 1135.47021
[3] S. Stević, “Generalized composition operators from logarithmic Bloch spaces to mixed-norm spaces,” Utilitas Mathematica, vol. 77, pp. 167-172, 2008. · Zbl 1175.47033
[4] S. Stević, “On an integral operator between Bloch-type spaces on the unit ball,” Bulletin des Sciences Mathematiques. In press. · Zbl 1189.47032
[5] S. Stević, “Products of integral-type operators and composition operators from a mixed norm space to Bloch-type spaces,” Siberian Mathematical Journal, vol. 50, no. 4, pp. 726–736, 2009. · Zbl 1219.47050
[6] X. Zhu, “Generalized composition operators from generalized weighted Bergman spaces to Bloch type spaces,” Journal of the Korean Mathematical Society, vol. 46, no. 6, pp. 1219-1232, 2009. · Zbl 1197.47042
[7] S. Stević and S.-I. Ueki, “Integral-type operators acting between weighted-type spaces on the unit ball,” Applied Mathematics and Computation, vol. 215, no. 7, pp. 2464-2471, 2009. · Zbl 1197.47061
[8] D.-C. Chang, S. Li, and S. Stević, “On some integral operators on the unit polydisk and the unit ball,” Taiwanese Journal of Mathematics, vol. 11, no. 5, pp. 1251-1285, 2007. · Zbl 1149.47026
[9] Z. Hu, “Extended Cesàro operators on mixed norm spaces,” Proceedings of the American Mathematical Society, vol. 131, no. 7, pp. 2171-2179, 2003. · Zbl 1054.47023
[10] S. G. Krantz and S. Stević, “On the iterated logarithmic Bloch space on the unit ball,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 5-6, pp. 1772-1795, 2009. · Zbl 1221.47056
[11] S. Stević, “On an integral operator on the unit ball in \Bbb Cn,” Journal of Inequalities and Applications, vol. 2005, no. 1, pp. 81-88, 2005. · Zbl 1074.47013
[12] S. Stević, “Boundedness and compactness of an integral operator on a weighted space on the polydisc,” Indian Journal of Pure and Applied Mathematics, vol. 37, no. 6, pp. 343-355, 2006. · Zbl 1121.47032
[13] S. Stevich, “Boundedness and compactness of an integral operator in a mixed norm space on the polydisk,” Sibirskii Matematicheskii Zhurnal, vol. 48, no. 3, pp. 694-706, 2007. · Zbl 1164.47331
[14] S. Stević, “On a new operator from H\infty to the Bloch-type space on the unit ball,” Utilitas Mathematica, vol. 77, pp. 257-263, 2008. · Zbl 1175.47034
[15] S. Stević, “On a new integral-type operator from the weighted Bergman space to the Bloch-type space on the unit ball,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 154263, 14 pages, 2008. · Zbl 1155.32002
[16] S. Stević, “On a new operator from the logarithmic Bloch space to the Bloch-type space on the unit ball,” Applied Mathematics and Computation, vol. 206, no. 1, pp. 313-320, 2008. · Zbl 1162.47029
[17] S. Stević, “On a new integral-type operator from the Bloch space to Bloch-type spaces on the unit ball,” Journal of Mathematical Analysis and Applications, vol. 354, no. 2, pp. 426-434, 2009. · Zbl 1171.47028
[18] S. Stević, “On an integral-type operator from logarithmic Bloch-type and mixed-norm spaces to Bloch-type spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 12, pp. 6323-6342, 2009. · Zbl 1186.47033
[19] J. Xiao, “Riemann-Stieltjes operators on weighted Bloch and Bergman spaces of the unit ball,” Journal of the London Mathematical Society, vol. 70, no. 1, pp. 199-214, 2004. · Zbl 1064.47034
[20] X. Tang, “Extended Cesàro operators between Bloch-type spaces in the unit ball of Cn,” Journal of Mathematical Analysis and Applications, vol. 326, no. 2, pp. 1199-1211, 2007. · Zbl 1117.47022
[21] W. Yang, “On an integral-type operator between Bloch-type spaces,” Applied Mathematics and Computation, vol. 215, no. 3, pp. 954-960, 2009. · Zbl 1185.45020
[22] S. Stević and E. Wolf, “Differences of composition operators between weighted-type spaces of holomorphic functions on the unit ball of \Bbb Cn,” Applied Mathematics and Computation, vol. 215, no. 5, pp. 1752-1760, 2009. · Zbl 1177.47033
[23] K. D. Bierstedt and W. H. Summers, “Biduals of weighted Banach spaces of analytic functions,” Journal of Australian Mathematical Society. Series A, vol. 54, no. 1, pp. 70-79, 1993. · Zbl 0801.46021
[24] 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
[25] K. D. Bierstedt, J. Bonet, and J. Taskinen, “Associated weights and spaces of holomorphic functions,” Studia Mathematica, vol. 127, no. 2, pp. 137-168, 1998. · Zbl 0934.46027
[26] M. Lindström and E. Wolf, “Essential norm of the difference of weighted composition operators,” Monatshefte für Mathematik, vol. 153, no. 2, pp. 133-143, 2008. · Zbl 1146.47015
[27] K. Madigan and A. Matheson, “Compact composition operators on the Bloch space,” Transactions of the American Mathematical Society, vol. 347, no. 7, pp. 2679-2687, 1995. · Zbl 0826.47023
[28] S. Stević and S.-I. Ueki, “Weighted composition operators and integral-type operators between weighted Hardy spaces on the unit ball,” Discrete Dynamics in Nature and Society, vol. 2009, Article ID 952831, 21 pages, 2009. · Zbl 1178.32002
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