Extended Cesàro operators between Bloch-type spaces in the unit ball of \(\mathbf {C}^{n}\). (English) Zbl 1117.47022

Let \(H(B)\) be the space of all holomorphic functions on \(B\), the unit ball of \(\mathbb C^n\). By an extended Cesàro operator with symbol \(\varphi\in H(B)\), often called a Volterra operator in the literature, the author means an operator \(T_\varphi\) on \(H(B)\) defined by \[ T_\varphi f(z) = \int_0^1 f(tz) {\mathcal R} \varphi(tz)\,{dt\over t}, \] where \({\mathcal R}\varphi\) denotes the radial derivative of \(\varphi\).
In this paper, the author studies the boundedness or compactness of \(T_\varphi\) acting between certain weighted (little) Bloch-type spaces \({\mathcal B}_\omega({\mathcal B}_{\omega, 0})\) where \(\omega\) is a weight (called “normal” by the author) satisfying some natural growth conditions near the boundary. For boundedness, the author proves that \(T_\varphi: {\mathcal B}_\omega ({\mathcal B}_{\omega, 0}) \to {\mathcal B}_\mu ({\mathcal B}_{\mu, 0})\) is bounded if and only if \(\sup\mu(z)| {\mathcal R}\varphi(z)| \int_0^{| z| }\omega^{-1}(t)\,dt<\infty\) (and \(\varphi\in {\mathcal B}_{\mu, 0}\)). For little Bloch-type spaces, the corresponding little oh condition is shown to be equivalent to the compactness of \(T_\varphi: {\mathcal B}_{\omega, 0} \to {\mathcal B}_{\mu, 0}\). More interestingly, such little oh characterization is no longer true for Bloch-type spaces in general. In fact, if \(\int_0^{1}\omega^{-1}(t)\,dt=\infty\), then the corresponding little oh condition is shown to be equivalent to the compactness of \(T_\varphi: {\mathcal B}_\omega \to {\mathcal B}_\mu\). However, if \(\int_0^{1}\omega^{-1}(t)dt<\infty\), then it is shown that \(T_\varphi: {\mathcal B}_\omega \to {\mathcal B}_\mu\) is compact if and only if \(\varphi\in {\mathcal B}_\mu\).
These results provide a unified way of treating several earlier results on Bloch-type spaces with standard weights.


47B38 Linear operators on function spaces (general)
32A36 Bergman spaces of functions in several complex variables
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[1] Miao, J., The Cesàro operator is bounded on \(H^p\) for \(0 < p < 1\), Proc. amer. math. soc., 116, 1077-1079, (1992) · Zbl 0787.47029
[2] Pommerenke, Ch., Schlichte funktionen und analytische funktionen von beschrankter mittlerer oszilation, Comment. math. helv., 52, 591-602, (1977) · Zbl 0369.30012
[3] Siskakis, A.G., Composition semigroups and the Cesàro operator on \(H^p\), J. London math. soc., 36, 2, 153-164, (1987) · Zbl 0634.47038
[4] Siskakis, A.G., On the Bergman space norm of the Cesàro operator, Arch. math., 67, 312-318, (1996) · Zbl 0859.47024
[5] Xiao, J., Cesàro operators on Hardy, BMOA and Bloch spaces, Arch. math., 68, 398-406, (1997) · Zbl 0870.30026
[6] Xiao, J.; Tan, H., p-Bergman spaces α-Bloch spaces little α-Bloch spaces and Cesàro means, Chinese ann. math. ser. A, 19, 2, 187-196, (1998), (in Chinese) · Zbl 0914.30021
[7] Aleman, A.; Siskakis, A.G., Integration operators on Bergman spaces, Indiana univ. math. J., 46, 337-356, (1997) · Zbl 0951.47039
[8] Aleman, A.; Cima, J., An integral operator on \(H^p\) and Hardy’s inequality, J. anal. math., 85, 157-176, (2001) · Zbl 1061.30025
[9] Wang, S.S.; Hu, Z.J., Extended Cesàro operators on Bloch-type spaces, Chinese ann. math. ser. A, 26, 5, 613-624, (2005), (in Chinese) · Zbl 1099.47507
[10] Hu, Z.J., Extended Cesàro operators on the Bloch space in the unit ball of \(\mathbf{C}^n\), Acta math. sci. ser. B, 23, 4, 561-566, (2003) · Zbl 1044.47023
[11] Xiao, J., Riemann – stieltjes operators on weighted Bloch and Bergman spaces of the unit ball, J. London math. soc. (2), 70, 199-214, (2004) · Zbl 1064.47034
[12] Stević, S., On integral operator on the unit ball in \(\mathbf{C}^n\), J. inequal. appl., 1, 81-88, (2005) · Zbl 1074.47013
[13] Zhang, X.J., Weighted Cesàro operators on Dirichlet type spaces and Bloch type spaces of \(\mathbf{C}^n\), Chinese ann. math. ser. A, 26, 1, 139-150, (2005), (in Chinese) · Zbl 1088.47504
[14] Hu, Z.J., Extended Cesàro operators on Bergman spaces, J. math. anal. appl., 296, 435-454, (2004) · Zbl 1072.47029
[15] Rudin, W., Function theory in the unit ball of \(\mathbf{C}^n\), (1980), Springer-Verlag New York
[16] Hu, Z.J.; Wang, S.S., Composition operators on Bloch-type spaces, Proc. roy. soc. Edinburgh sect. A, 135, 1229-1239, (2005) · Zbl 1131.47019
[17] Madigan, K.; Matheson, A., Compact composition operators on the Bloch space, Trans. amer. math. soc., 347, 2679-2687, (1995) · Zbl 0826.47023
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