## 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.

### MSC:

 47B38 Linear operators on function spaces (general) 32A36 Bergman spaces of functions in several complex variables

### Keywords:

Bloch-type space; extended Cesàro operator; unit ball
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### References:

 [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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