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

Tang, Xiaomin

J. Math. Anal. Appl. 326, No. 2, 1199-1211 (2007).

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.

Reviewer: Boo Rim Choe (Seoul)

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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