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A sharp blow-up estimate for the Lebesgue norm

  • Fernando Farroni
  • Alberto FiorenzaEmail author
  • Raffaella Giova
Article
  • 29 Downloads

Abstract

We prove that if \(p>1\) and \(\psi :]0,p-1[\rightarrow ]0,\infty [\) is nondecreasing, then
$$\begin{aligned} \sup _{0<\varepsilon<p-1} \psi (\varepsilon ) \Vert f\Vert _{L^{p-\varepsilon }(0,1)}\approx & {} \sup _{0<t<1} \psi \left( \frac{p-1}{1-\log t}\right) \Vert f^*\Vert _{L^{p}(t,1)} \\&\mathop {\Updownarrow }\limits _{{\begin{array}{c} \psi \,\in \,\Delta _2\,\cap \, L^\infty . \end{array}}} \end{aligned}$$
Here f is a Lebesgue measurable function on (0, 1) and \(f^*\) denotes the decreasing rearrangement of f. The proof generalizes and makes sharp an equivalence previously known only in the particular case when \(\psi \) is a power; such case had a relevant role for the study of grand Lebesgue spaces. A number of consequences are discussed, among which: the behavior of the fundamental function of generalized grand Lebesgue spaces, an analogous equivalence in the case the assumption on the monotonicity of \(\psi \) is dropped, and an optimal estimate of the blow-up of the Lebesgue norms for functions in Orlicz–Zygmund spaces.

Keywords

Lebesgue spaces Grand Lebesgue spaces Euler’s Gamma function Orlicz–Zygmund spaces Fundamental function \(\Delta _2\) Condition Norm blow-up Banach function spaces 

Mathematics Subject Classification

46E30 26D15 

Notes

Acknowledgements

The research of the first and the third author has been partially supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The third author acknowledges also the support by Università degli Studi di Napoli Parthenope through the Project “Sostegno alla ricerca individuale (annualità 2015–2016–2017)” and the Project “Sostenibilità, esternalità e uso efficiente delle risorse ambientali (triennio 2017–2019)”. The authors are grateful to the anonymous referees for the very detailed reports, rich of pertinent issues. Because of their several comments, the exposition has been significantly improved.

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Copyright information

© Universidad Complutense de Madrid 2019

Authors and Affiliations

  • Fernando Farroni
    • 1
  • Alberto Fiorenza
    • 2
    • 3
    Email author
  • Raffaella Giova
    • 4
  1. 1.Dipartimento di Matematica e Applicazioni R. CaccioppoliUniversità di Napoli Federico IINapoliItaly
  2. 2.Dipartimento di ArchitetturaUniversità di Napoli Federico IINapoliItaly
  3. 3.Istituto per le Applicazioni del Calcolo “Mauro Picone”, sezione di NapoliConsiglio Nazionale delle RicercheNapoliItaly
  4. 4.Università degli Studi di Napoli ParthenopeNapoliItaly

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