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Weighted composition operators from Bergman-type spaces into Bloch spaces. (English) Zbl 1130.47016
It is a quite common trend in the study of composition operators between spaces of analytic functions in the unit disc of the complex plane that if boundedness corresponds to a “big-oh” condition, then compactness is determined by the “little-oh” of such condition; see, for instance, Chapter 3 in [J. H. Shapiro, “Composition operators and classical function theory” (Universitext: Tracts in Mathematics; New York: Springer–Verlag (1993; Zbl 0791.30033)]. The present article follows this trend. Given \(0<p<+\infty\) and a continuous function \(\phi:[0,1)\to (0,+\infty)\) such that for some \(0<s<t,\) \(\phi(r)=o(1-r)^s\) and \((1-r)^t=o(\phi(r))\) as \(r\to 1,\) the weighted Bergman-type space \(H(p,p,\phi)\) is the space of all analytic functions \(f\) in the unit disc such that \(\| f\| _{p,\phi}:=\int_{\mathbb D}| f(z)| ^p {\phi^p(| z| )\over{1-| z| }}\, dA(z)\) is finite, where \(dA\) is the normalized Lebesgue measure in the unit disc. The authors prove that a weighted composition operator \(uC_\varphi\) acting from \(H(p,p,\phi)\) into the Bloch space \({\mathcal B}\) is bounded if and only if \[ (1-| z| ^2)| u'(z)| =O\left(\phi(| \varphi(z)| )(1-| \varphi(z)| ^2)^{1/ p}\right) \text{ and } \] \[ \;(1-| z| ^2)| u(z)\varphi'(z)| =O\left(\phi(| \varphi(z)| )(1-| \varphi(z)| ^2)^{1+1/ p}\right). \] The “little-oh” of such condition when \(| \varphi(z)| \to 1,\) respectively when \(| z| \to 1,\) characterizes the compactness of \(uC_\varphi: H(p,p,\phi)\to {\mathcal B},\) respectively \(uC_\varphi: H(p,p,\phi)\to {\mathcal B}_0,\) the little Bloch space. The subsequent corollaries for the Bergman spaces \(A^p=H(p,p,(1-r)^{1/p})\) are stated.

MSC:
47B33 Linear composition operators
46E15 Banach spaces of continuous, differentiable or analytic functions
30H05 Spaces of bounded analytic functions of one complex variable
30D45 Normal functions of one complex variable, normal families
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