Superposition operators between the Bloch space and Bergman spaces (Q1880450)

The purpose of this article is to characterize the entire functions \(\varphi\) which transform by superposition, \(S_{\varphi}:f \rightarrow \varphi \circ f\), the Bloch space \(B\) into the Bergman space \(A^p\), \(0\ll \infty\). First of all they show that \(S_{\varphi}\) maps \(A^p\) into \(B\) if and only if \(\varphi\) is constant. The main result shows that \(S_{\varphi}\) maps \(B\) into \(A^p\) if and only if \(S_{\varphi}\) maps bounded sets in \(B\) into bounded sets in \(A^p\), and if and only if \(\varphi\) is an entire function either of order less than one or of order one and type zero. The key point is the geometric construction of a special univalent function. This function is a conformal map onto a specific Bloch domain with the maximal logarithmic growth along a certain polygonal line. Similar constructions of simply connected domains as images of functions in certain function spaces had already appeared in \textit{S. M. Buckley, J. L. Fernández}, \textit{D. Vukotić} [Papers on Analysis: A volume dedicated to Olli Martio, Rep. Univ. Jyväskylä Dept. Math. Stat., Jyväskylä 83, 41--61 (2001; Zbl 1032.30031)], and \textit{J. J. Donaire, D. Girela} and \textit{D. Vukotić}, [J. Reine Angew. Math. 553, 43--72 (2002; Zbl 1006.30031)]. At the end of the article it is observed that similar results hold if one replaces the Bloch space \(B\) by the space BMOA.