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On functions having coincident p-norms

  • Giuliano KlunEmail author
Article
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Abstract

In a measure space \((X,{\mathcal {A}},\mu )\), we consider two measurable functions \(f,g:E\rightarrow {\mathbb {R}}\), for some \(E\in {\mathcal {A}}\). We prove that the property of having equal p-norms when p varies in some infinite set \(P\subseteq [1,+\infty )\) is equivalent to the following condition:
$$\begin{aligned} \mu (\{x\in E:|f(x)|>\alpha \})=\mu (\{x\in E:|g(x)|>\alpha \})\quad \text { for all } \alpha \ge 0. \end{aligned}$$

Keywords

Lebesgue integrable functions \({\mathcal {L}}^p\)-norms Mellin transform 

Mathematics Subject Classification

28A25 

Notes

Acknowledgements

The author wishes to warmly thank Giovanni Alessandrini for his precious help, and Gianni Dal Maso, Daniele Del Santo, Alessandro Fonda and Sergio Vesnaver for the useful discussions and suggestions. I am also very grateful to the anonymous referee for pointing out some gaps in the proofs and suggesting several improvements.

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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Scuola Internazionale Superiore di Studi AvanzatiTriesteItaly

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