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

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