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Superposition operators between mixed norm spaces of analytic functions

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Abstract

If \(\varphi \) is an entire function, the superposition operator \(S_\varphi \) is defined in the space \({\mathcal {H}}{\mathcal {o}}{\mathcal {l}}({\mathbb {D}})\) of all analytic functions f in the unit disc \({\mathbb {D}}\), by \(S_{\varphi }(f)=\varphi \circ f\). We consider the mixed norm spaces of Hardy type \(H(p,q,\alpha )\) (\(0<p,q\le \infty \), \(\alpha >0\)). In this work we provide a complete characterization of those entire functions \(\varphi \) so that the superposition operator \(S_\varphi \) maps \(H(p,q,\alpha )\) into \(H(s,t,\beta )\) for any two triplets of admissible parameters \((p,q,\alpha )\) and \((s,t,\beta )\). We also prove that superposition operators mapping a mixed norm space into another are bounded and continuous.

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Acknowledgements

We wish to thank the referee for reading carefully the paper and making a number of nice suggestions to improve it.

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  1. Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071, Málaga, Spain

    Salvador Domínguez & Daniel Girela

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  1. Salvador Domínguez
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  2. Daniel Girela
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Correspondence to Daniel Girela.

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This research has been supported in part by a grant from “El Ministerio de Ciencia, Innovación y Universidades” , Spain (PGC2018-096166-B-I00) and by Grants from la Junta de Andalucía (FQM-210 and UMA18-FEDERJA-002).

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Domínguez, S., Girela, D. Superposition operators between mixed norm spaces of analytic functions. Mediterr. J. Math. 18, 18 (2021). https://doi.org/10.1007/s00009-020-01667-4

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