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Composition operators between Hardy and Bloch-type spaces of the upper half-plane. (English) Zbl 1142.47019
Let \(H^p(\pi^+)\) be the Hardy space of those functions \(f\) holomorphic on the upper half plane \(\pi^+\) (say, \(f\in H(\pi^+)\)) for which \[ \| f\|^p_p=\sup_{y>0}\int_{-\infty}^\infty| f(x+iy)|^p dx<\infty. \] The Bloch space \(B_\infty(\pi^+)\) of the upper half-plane is the set of all \(f\in H(\pi^+)\) satisfying \(| | f| |_{B_\infty}=\sup_{z\in \pi^+}\text{ Im}\; z \; | f '(z)| <\infty\). Finally, a function \(f\in H(\pi^+)\) is said to belong to \(\mathcal A_\infty(\pi^+)\) if \(\| f\|_{\mathcal A_\infty}=\sup_{z\in\pi^+}\text{Im}\, z \, | f (z)| <\infty\). The authors give necessary and sufficient conditions on holomorphic selfmaps \(\phi\) of \(\pi^+\) in order the composition operators \(C_\phi: f\mapsto f\circ\phi\) are bounded acting as operators on \(\mathcal A_\infty(\pi^+)\), respectively, as operators between \(H^p(\pi^+)\) and \(\mathcal A_\infty(\pi^+)\) or \(H^p(\pi^+)\) and \(B_\infty(\pi^+)\).

47B33 Linear composition operators
46E10 Topological linear spaces of continuous, differentiable or analytic functions
30D55 \(H^p\)-classes (MSC2000)
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