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Integral-type operators acting between weighted-type spaces on the unit ball. (English) Zbl 1197.47061

Let \(\mathbb{B}\) denote the unit ball in \(\mathbb{C}^n\) and \(H(\mathbb{B})\) the space of all holomorphic functions on \(\mathbb{B}\). In this paper, the authors study the boundedness and compactness of the integral-type operators
\[ I_\varphi^gf(z)=\int_0^1 \operatorname{Re} f(\varphi(tz))g(tz)\frac{dt}{t},\quad z\in \mathbb B, \]
where \(g\in H(\mathbb{B})\), \(g(0)=0\), \(\varphi\) is a holomorphic self-map of \(\mathbb{B}\) and \(\operatorname{Re} f\) is the radial derivative of the function \(f\), between weighted-type spaces on the unit ball.

MSC:

47G10 Integral operators
47B38 Linear operators on function spaces (general)
32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))
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[1] Avetisyan, K.; Stević, S., Generalized Libera transform is bounded on Besov mixed-norm, BMOA and VMOA spaces on the unit disk, Appl. Math. Comput., 213, 304-311 (2009) · Zbl 1167.30325
[2] Avetisyan, K.; Stević, S., Extended Cesàro operators between different Hardy spaces, Appl. Math. Comput., 207, 346-350 (2009) · Zbl 1163.32004
[3] Chang, D. C.; Li, S.; Stević, S., On some integral operators on the unit polydisk and the unit ball, Taiwanese J. Math., 11, 5, 1251-1286 (2007) · Zbl 1149.47026
[4] D. Clahane, S. Stević, Norm equivalence and composition operators between Bloch/Lipschitz spaces of the unit ball, J. Inequal. Appl. vol. 2006, Article ID 61018 (2006) 11 p.; D. Clahane, S. Stević, Norm equivalence and composition operators between Bloch/Lipschitz spaces of the unit ball, J. Inequal. Appl. vol. 2006, Article ID 61018 (2006) 11 p. · Zbl 1131.47018
[5] D. Gu, Extended Cesàro operators from logarithmic-type spaces to Bloch-type spaces, Abstr. Appl. Anal. vol. 2009, Article ID 246521 (2009) 9 p.; D. Gu, Extended Cesàro operators from logarithmic-type spaces to Bloch-type spaces, Abstr. Appl. Anal. vol. 2009, Article ID 246521 (2009) 9 p.
[6] Hu, Z. J., Extended Cesàro operators on mixed norm spaces, Proc. Am. Math. Soc., 131, 7, 2171-2179 (2003) · Zbl 1054.47023
[7] Krantz, S.; Stević, S., On the iterated logarithmic Bloch space on the unit ball, Nonlinear Anal. TMA, 71, 1772-1795 (2009) · Zbl 1221.47056
[8] Li, S., Generalized Hilbert operator on the Dirichlet-type space, Appl. Math. Comput., 214, 1, 304-309 (2009) · Zbl 1167.47031
[9] Li, S.; Stević, S., Integral type operators from mixed-norm spaces to \(α\)-Bloch spaces, Integral Transform. Spec. Funct., 18, 7, 485-493 (2007) · Zbl 1131.47031
[10] Li, S.; Stević, S., Riemann-Stieltjes operators on Hardy spaces in the unit ball of \(C^n\), Bull. Belg. Math. Soc. Simon Stevin, 14, 621-628 (2007) · Zbl 1136.47023
[11] Li, S.; Stević, S., Riemann-Stieltjes type integral operators on the unit ball in \(C^n\), Complex Variables Elliptic Equat., 52, 6, 495-517 (2007) · Zbl 1124.47022
[12] Li, S.; Stević, S., Compactness of Riemann-Stieltjes operators between \(F(p, q, s)\) and \(α\)-Bloch spaces, Publ. Math. Debrecen, 72, 1-2, 111-128 (2008) · Zbl 1164.47040
[13] Li, S.; Stević, S., Generalized composition operators on Zygmund spaces and Bloch-type spaces, J. Math. Anal. Appl., 338, 1282-1295 (2008) · Zbl 1135.47021
[14] Li, S.; Stević, S., Products of composition and integral type operators from \(H^\infty\) to the Bloch space, Complex Variables Elliptic Equat., 53, 5, 463-474 (2008) · Zbl 1159.47019
[15] Li, S.; Stević, S., Riemann-Stieltjes operators between different weighted Bergman spaces, Bull. Belg. Math. Soc. Simon Stevin, 15, 4, 677-686 (2008) · Zbl 1169.47026
[16] Li, S.; Stević, S., Riemann-Stieltjes operators between mixed norm spaces, Indian J. Math., 50, 1, 177-188 (2008) · Zbl 1159.47012
[17] Li, S.; Stević, S., Products of Volterra type operator and composition operator from \(H^\infty\) and Bloch spaces to the Zygmund space, J. Math. Anal. Appl., 345, 40-52 (2008) · Zbl 1145.47022
[18] Li, S.; Stević, S., Cesàro type operators on some spaces of analytic functions on the unit ball, Appl. Math. Comput., 208, 378-388 (2009) · Zbl 1166.45009
[19] Li, S.; Stević, S., Integral-type operators from Bloch-type spaces to Zygmund-type spaces, Appl. Math. Comput., 215, 464-473 (2009) · Zbl 1179.45022
[20] Li, S.; Stević, S., Products of integral-type operators and composition operators between Bloch-type spaces, J. Math. Anal. Appl., 349, 596-610 (2009) · Zbl 1155.47036
[21] Shields, A. L.; Williams, D. L., Bounded projections, duality, and multipliers in spaces of analytic functions, Trans. Am. Math. Soc., 162, 287-302 (1971) · Zbl 0227.46034
[22] Stević, S., On an integral operator on the unit ball in \(C^n\), J. Inequal. Appl., 2005, 1, 81-88 (2005) · Zbl 1074.47013
[23] Stević, S., Boundedness and compactness of an integral operator on a weighted space on the polydisc, Indian J. Pure Appl. Math., 37, 6, 343-355 (2006) · Zbl 1121.47032
[24] Stević, S., Boundedness and compactness of an integral operator on mixed norm spaces on the polydisc, Siberian Math. J., 48, 3, 559-569 (2007)
[25] Stević, S., Norm of weighted composition operators from Bloch space to \(H_\mu^\infty\) on the unit ball, Ars. Combin., 88, 125-127 (2008) · Zbl 1224.30195
[26] S. Stević, On a new integral-type operator from the weighted Bergman space to the Bloch-type space on the unit ball, Discrete Dyn. Nat. Soc. vol. 2008, Article ID 154263 (2008) 14 p.; S. Stević, On a new integral-type operator from the weighted Bergman space to the Bloch-type space on the unit ball, Discrete Dyn. Nat. Soc. vol. 2008, Article ID 154263 (2008) 14 p.
[27] Stević, S., On a new operator from \(H^\infty\) to the Bloch-type space on the unit ball, Util. Math., 77, 257-263 (2008) · Zbl 1175.47034
[28] Stević, S., On a new operator from the logarithmic Bloch space to the Bloch-type space on the unit ball, Appl. Math. Comput., 206, 313-320 (2008) · Zbl 1162.47029
[29] Stević, S., On a new integral-type operator from the Bloch space to Bloch-type spaces on the unit ball, J. Math. Anal. Appl., 354, 426-434 (2009) · Zbl 1171.47028
[30] Stević, S., On an integral operator from the Zygmund space to the Bloch-type space on the unit ball, Glasg. J. Math., 51, 275-287 (2009) · Zbl 1176.47029
[31] Stević, S., Products of integral-type operators and composition operators from the mixed norm space to Bloch-type spaces, Siberian Math. J., 50, 4, 726-736 (2009)
[32] S. Stević, Integral-type operators from the mixed-norm space to the Bloch-type space on the unit ball, Siberian Math. J. (in press).; S. Stević, Integral-type operators from the mixed-norm space to the Bloch-type space on the unit ball, Siberian Math. J. (in press). · Zbl 1217.47069
[33] S. Stević, S.I. Ueki, Weighted composition operators and integral-type operators between weighted Hardy spaces on the unit ball, Discrete Dyn. Nat. Soc. vol. 2009, Article ID 952831 (2009) 20 p.; S. Stević, S.I. Ueki, Weighted composition operators and integral-type operators between weighted Hardy spaces on the unit ball, Discrete Dyn. Nat. Soc. vol. 2009, Article ID 952831 (2009) 20 p.
[34] Yang, W., On an integral-type operator between Bloch-type spaces, Appl. Math. Comput., 215, 954-960 (2009) · Zbl 1185.45020
[35] Y. Yu, Y. Liu, On Li-Stević integral-type operators between different weighted Bloch-type spaces, J. Inequal. Appl. vol. 2008, Article ID 780845 (2008) 14 p.; Y. Yu, Y. Liu, On Li-Stević integral-type operators between different weighted Bloch-type spaces, J. Inequal. Appl. vol. 2008, Article ID 780845 (2008) 14 p.
[36] Zhu, X., Products of differentiation, composition and multiplication from Bergman-type spaces to Bers spaces, Integral Transform. Spec. Funct., 18, 3, 223-231 (2007) · Zbl 1119.47035
[37] Zhu, X., Extended Cesáro operator from \(H^\infty\) to Zygmund type spaces in the unit ball, J. Comput. Anal. Appl., 11, 2, 356-363 (2009) · Zbl 1221.47066
[38] X. Zhu, Generalized composition operators from generalized weighted Bergman spaces to Bloch-type spaces, J. Korean Math. Soc. (in press).; X. Zhu, Generalized composition operators from generalized weighted Bergman spaces to Bloch-type spaces, J. Korean Math. Soc. (in press). · Zbl 1197.47042
[39] Zhu, X., Weighted composition operators from logarithmic Bloch spaces to a class of weighted-type spaces in the unit ball, Ars. Combin., 91, 87-95 (2009) · Zbl 1216.47043
[40] Zhu, X., Integral-type operators from iterated logarithmic Bloch spaces to Zygmund-type spaces, Appl. Math. Comput., 215, 1170-1175 (2009) · Zbl 1185.45021
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