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Riemann-Stieltjes-type integral operators on the unit ball in $$\mathbb C^n$$. (English) Zbl 1124.47022
The authors investigate the boundedness and compactness of the operators $T_g(f)(z)=\int_0^1f(tz) {\mathcal R}g(tz)\frac{dt}{t}\,$ and $I_g(f)(z)=\int_0^1 {\mathcal R}f(tz)f(tz)\,\frac{dt}{t},$ where $$f$$ and $$g$$ are holomorphic functions on the unit ball of $${\mathbb C^n}$$ and $${\mathcal R}g$$ stands for the radial derivative $${\mathcal R}h(z)=\sum_{j=1}^n z_j \frac{\partial h}{\partial z_j}(z)$$ on Bloch-type spaces $${\mathcal B}^\alpha$$ (resp., $${\mathcal B}_0^\alpha$$) given by holomorphic functions $$f$$ satisfying $$\sup(1-| z| ^2)^\alpha| {\mathcal R}f(z)| <\infty$$ (resp., $$\lim_{| z| \to 1}(1-| z| ^2)^\alpha| {\mathcal R}f(z)| =0$$). Among other things, they show that, for $$0<\alpha<1$$, any bounded operator $$T_g$$ acting from $${\mathcal B}^\alpha$$ into $${\mathcal B}^\beta$$ is compact and this fact is equivalent to $$g\in {\mathcal B}^\beta$$. In the case $$\alpha=1$$ (resp., $$\alpha>1$$), the boundedness of from $$T_g$$ acting from $${\mathcal B}^\alpha$$ into $${\mathcal B}^\beta$$ is characterized by $$\sup(1-| z| ^2)^\beta\log(\frac{1}{1-| z| ^2})| {\mathcal R}f(z)| <\infty$$ (resp., $$\sup(1-| z| ^2)^{\beta-\alpha+1}| {\mathcal R}f(z)| <\infty$$), while compactness is equivalent to weak-compactness and is described by $$\lim_{| z| \to 1}(1-| z| ^2)^\beta\log(\frac{1}{1-| z| ^2})| {\mathcal R}f(z)| =0$$ (resp., $$\lim_{| z| \to 1}(1-| z| ^2)^{\beta-\alpha+1}| {\mathcal R}f(z)| =0).$$ The situation for the operator $$I_g$$ is as follows: For $$\alpha>\beta$$, there is no $$g\neq 0$$ such that $$I_g$$ is bounded from $${\mathcal B}^\alpha$$ into $${\mathcal B}^\beta$$ and for $$\alpha\leq \beta$$, the boundedness is characterized by $$\sup (1-| z| ^2)^{\beta-\alpha}| g(z)| <\infty$$. The compactness is again equivalent to weak-compactness and is described, for $$\alpha\leq \beta$$, by $$\lim_{| z| \to 1} (1-| z| ^2)^{\beta-\alpha}| g(z)| =0$$ in this case. Some result about products of $$I_h$$ and $$T_g$$ are also provided.

##### MSC:
 47B38 Linear operators on function spaces (general) 46E15 Banach spaces of continuous, differentiable or analytic functions 47G10 Integral operators
##### Keywords:
Riemann-Stieltjes-type integral operators
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##### References:
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