## 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

### Keywords:

Cesàro operator; Bloch space; integral operator
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