## Extended Cesàro operators on Bergman spaces.(English)Zbl 1072.47029

Denote by $$H(B)$$ the class of all holomorphic functions on the open unit ball $$B$$ of $$\mathbb{C}^n$$ and, for $$g\in H(B)$$, let $$\operatorname{Re} g(z)=\sum_{j=1}^n z_j (\partial g/\partial z_j)$$ denote the radial derivative. For $$g\in H(B)$$, the extended Cesàro operator $$T_g$$ with symbol $$g$$ is the operator on $$H(B)$$ defined by $T_g(f)(z)=\int_0^1 f(tz)\operatorname{Re} g(tz) \frac{dt}{t},\quad f\in H(B),\; z\in B.$ For a positive Lebesgue measurable function $$w$$ defined on $$B$$ and for $$0<p<\infty$$, the weighted Bergman space $$L_{a,w}^p(B)$$ is the space of all functions $$f\in H(B)$$ such that $\int_B | f(z)| ^pw(z)dm(z)<\infty.$ The main result of the paper under review characterizes (for some class of weights $$w$$) all symbols $$g$$ for which $$T_g$$ is a bounded operator from the Bergman space $$L_{a,w}^p(B)$$ to $$L_{a,w}^q(B)$$ for $$0<p, q<\infty$$. A similar characterization is given for compact operators.

### MSC:

 47B38 Linear operators on function spaces (general) 46E15 Banach spaces of continuous, differentiable or analytic functions 32A36 Bergman spaces of functions in several complex variables
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### References:

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