On an integral operator on the unit ball in \(\mathbb{C}^n\). (English) Zbl 1074.47013

Stević, Stevo

J. Inequal. Appl. 2005, No. 1, 81-88 (2005).

Let \(H(B)\) denote the space of all holomorphic functions on the unit ball \(B\subset \mathbb{C}^n\). In this paper, the integral operator \[ T_g(f)(z)=\int^1_0f(tz)R g(tz)(dt/t) \] is investigated, where \(f\in H(B), z\in B\), \(g\in H(B)\) and \(Rg(z)=\sum\limits^n_{j=1}z_j(\partial g){\partial z_j}(z)\) is the radial derivative of \(g\). This operator can be considered as an extension of the Cesàro operator on the unit disk. The present article characterizes those \(g\) for which \(T_g\) is bounded on \(\alpha\)-Bloch spaces.

Reviewer: Zehua Zhou (Tianjin)

MSC:

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

Keywords:

Cesàro operator; Bloch space; integral operator