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Riemann–Stieltjes operators from \(F(p,q,s)\) spaces to \(\alpha\)-Bloch spaces on the unit ball. (English) Zbl 1131.47030

Summary: Let \(H(B)\) denote the space of all holomorphic functions on the unit ball \(B\subset\mathbb C^n\). We investigate the following integral operators: \[ \begin{aligned} T_g(f)(z)&= \int_0^1 f(tz)\operatorname{Re}\,g(tz) (dt/t),\\ L_g(f)(z)&= \int_0^1 \operatorname{Re}\,f(tz)g(tz)(dt/t), \end{aligned} \] \(f\in H(B)\), \(z\in B\), where \(g\in H(B)\), and \(\operatorname{Re}\,h(z)= \sum_{j=1}^n z_j(\partial h/\partial z_j)(z)\) is the radial derivative of \(h\). The operator \(T_g\) can be considered as an extension of the Cesàro operator on the unit disk. The boundedness of two classes of Riemann–Stieltjes operators from general function space \(F(p,q,s)\), which includes Hardy space, Bergman space, \(Q_p\) space, BMOA space, and Bloch space, to \(\alpha\)-Bloch space \({\mathcal B}^\alpha\) in the unit ball is discussed in this paper.

MSC:

47B38 Linear operators on function spaces (general)
32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))
46E15 Banach spaces of continuous, differentiable or analytic functions
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