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.


47B38 Linear operators on function spaces (general)
46E15 Banach spaces of continuous, differentiable or analytic functions
47G10 Integral operators
Full Text: DOI


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