## Composition operators between generally weighted Bloch spaces and $$Q_{\log}^q$$ space.(English)Zbl 1163.47019

Let $$H(\mathbb D)$$ be the set of all holomorphic functions $$f$$ in the open unit disk $$\mathbb D$$. For $$p,q>0$$, the weighted Bloch space $$B_{\log}^p$$ is the set of all functions in $$H(\mathbb D)$$ for which
$\| f\| _{B_{\log}^p} =| f(0)| +\sup_{z\in \mathbb D}| f '(z)| (1-| z| ^2)^p\log\frac{2}{1-| z| ^2}<\infty,$
and $$Q^q_{\log}$$ is the space of all $$f\in H(\mathbb D)$$ for which
$\| f\| _*=\sup_{I\subseteq\partial\mathbb D}\frac{\left(\log\frac{2}{| I| }\right)^2}{| I| ^q} \int_{S(I)}| f '(z)| ^2 \left(\log\frac{1}{| z| }\right)^q \,dm(z)<\infty,$
where $$dm$$ is planar Lebesgue measure. Here, as usual, $$S(I)$$ is the Carleson square $$\{z\in\mathbb D: 1-| I| \leq | z| <1, \frac{z}{| z| }\in I\}$$ associated with the arc $$I\subseteq\partial\mathbb D$$.
For analytic selfmaps $$\varphi$$ of $$\mathbb D$$, 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 $$f\in B_{\log}^p$$.

### MSC:

 47B33 Linear composition operators 47B38 Linear operators on function spaces (general) 30H05 Spaces of bounded analytic functions of one complex variable
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