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

Li, Haiying; Liu, Peide

Banach J. Math. Anal. 3, No. 1, 99-110 (2009).

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 \(f\in B_{\log}^p\).

Reviewer: Raymond Mortini (Metz)

MSC:

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

Keywords:

composition operators; weighted Bloch spaces; boundedness; compactness