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Zbl 1163.47022
Zhu, Xiangling
(Zhu, Xiang-ling)
Volterra composition operators on logarithmic Bloch spaces. (English)
[J] Banach J. Math. Anal. 3, No. 1, 122-130, electronic only (2009). ISSN 1735-8787/e

Let $\Bbb B$ be the unit ball in $\Bbb C^n$ and $H(\Bbb B)$ the set of all holomorphic functions on $\Bbb B$. Let ${\frak R} f(z)=\sum_{j=1}^n z_j \frac{\partial f}{\partial z_j} (z)$ be the radial derivative of $f\in H(\Bbb B)$ and $$A_f(z):=(1-\vert z\vert ^2)\left( \log\frac{e}{1-\vert z\vert ^2} \right) \vert {\frak R} f(z)\vert .$$ The logarithmic Bloch space $LB$ is the set of all functions $f\in H(\Bbb B)$ for which $ \beta(f):=\sup_{z\in \Bbb B}A_f(z)<\infty,$ and its little sibling is given by $$ \text{LB}_0= \Big\{f\in \text{LB}: \lim_{\vert z\vert \to1} A_f(z)=0\Big\}.$$ Let $T_{g,\varphi}$ be defined as follows: $$T_{g,\varphi}f(z)=\int_0^1 f(\varphi(tz)){\frak R}g(tz)\,\frac{dt}{t}.$$ Here, $g\in H(\Bbb B)$ and $\varphi$ is a holomorphic selfmap of $\Bbb B$. The author characterizes boundedness and compactness of these operators between the logarithmic Bloch spaces LB and $\text{LB}_0$.
[Raymond Mortini (Metz)]

MSC 2000:: *47B33 Composition operators
47B38 Operators on function spaces
32A37 Spaces of holomorphic functions (several variables)

Keywords: composition operators; logarithmic Bloch spaces on the ball; boundedness; compactness

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