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Composition operators from the Hardy space to the Zygmund-type space on the upper half-plane. (English) Zbl 1173.30036

Authors’ abstract: Here we introduce an \(n\)th weighted space on the upper half-plane \(\Pi _{+}=\{z\in \mathbb C:\text{Im}\, z>0\}\) in the complex plane \(\mathbb C\). For the case \(n=2\), we call it the Zygmund-type space, and denote it by \(\mathcal Z(\Pi _{+})\). The main result of the paper gives some necessary and sufficient conditions for the boundedness of the composition operator \(C_{\varphi }f(z)=f(\varphi (z))\) from the Hardy space \(H^{p}(\Pi _{+})\) on the upper half-plane, to the Zygmund-type space, where \(\varphi \) is an analytic self-map of the upper half-plane.

MSC:

30H05 Spaces of bounded analytic functions of one complex variable
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References:

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