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Zbl 1163.47019
Li, Haiying; Liu, Peide
(Li, Hai-ying; Liu, Pei-de)
Composition operators between generally weighted Bloch spaces and $Q_{\log}^q$ space.
(English)
[J] Banach J. Math. Anal. 3, No. 1, 99-110, electronic only (2009). ISSN 1735-8787/e

Let $H(\Bbb D)$ be the set of all holomorphic functions $f$ in the open unit disk $\Bbb D$. For $p,q>0$, the weighted Bloch space $B_{\log}^p$ is the set of all functions in $H(\Bbb D)$ for which $$\| f\| _{B_{\log}^p} =\vert f(0)\vert +\sup_{z\in \Bbb D}\vert f '(z)\vert (1-\vert z\vert ^2)^p\log\frac{2}{1-\vert z\vert ^2}<\infty,$$ and $Q^q_{\log}$ is the space of all $f\in H(\Bbb D)$ for which $$\| f\| _*=\sup_{I\subseteq\partial\Bbb D}\frac{\left(\log\frac{2}{\vert I\vert }\right)^2}{\vert I\vert ^q} \int_{S(I)}\vert f '(z)\vert ^2 \left(\log\frac{1}{\vert z\vert }\right)^q \,dm(z)<\infty,$$ where $dm$ is planar Lebesgue measure. Here, as usual, $S(I)$ is the Carleson square $\{z\in\Bbb D: 1-\vert I\vert \leq \vert z\vert <1, \frac{z}{\vert z\vert }\in I\}$ associated with the arc $I\subseteq\partial\Bbb D$. For analytic selfmaps $\varphi$ of $\Bbb D$, the author characterizes those composition operators $C_\varphi: B_{\log}^p\to Q^q_{\log}$, $f\mapsto f\circ \phi$, that are bounded, respectively compact. Also, necessary and sufficient conditions on the Taylor coefficients of a lacunary Taylor series $f$ are given that imply that $f\in B_{\log}^p$.
[Raymond Mortini (Metz)]
MSC 2000:
*47B33 Composition operators
47B38 Operators on function spaces
30H05 Spaces and algebras of analytic functions

Keywords: composition operators; weighted Bloch spaces; boundedness; compactness

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