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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 $$B_\infty(\pi^+)$$.

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