An estimate of the blow-up of Lebesgue norms in the non-tempered case

Giovanni Di Fratta, Alberto Fiorenza*, Valeriy Slastikov

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

2 Citations (Scopus)

Abstract

We prove that if p>1 and ψ:]0,p−1]→]0,∞[ is just nondecreasing and differentiable (hence not necessarily Δ2), then for every f Lebesgue measurable function on (0,1) sup0<ε<p−1⁡ψ(ε)‖f‖Lp−ε(0,1)≲sup0<t<1⁡Sψ(t)‖fLp(t,1), where f denotes the decreasing rearrangement of f and Sψ is defined, for ε∈]0,p−1[, through [Formula presented] where cψ is the normalizing constant chosen so that ν((p−1)−)=1. If ψ is in a class of functions satisfying the Δ2 condition, essentially characterized by the so-called ∇ condition, then inequality (⁎) is sharp, in the sense that both sides are equivalent. Estimate (⁎) generalizes an inequality of the type obtained by the second author with Farroni and Giova in [6] under the growth condition ψ∈Δ2.

Original languageEnglish
Article number124550
JournalJournal of Mathematical Analysis and Applications
Volume493
Issue number2
DOIs
Publication statusPublished - 15 Jan 2021

Bibliographical note

Funding Information:
The work of the third author was supported by the EPSRC grant EP/K02390X/1 and the Leverhulme grant RPG-2018-438 .

Funding Information:
The first author acknowledges support from the Austrian Science Fund (FWF) through the special research program Taming complexity in partial differential systems (Grant SFB F65 ).

Funding Information:
The first author acknowledges support from the Austrian Science Fund (FWF) through the special research program Taming complexity in partial differential systems (Grant SFB F65). All the authors acknowledge support from ESI, the Erwin Schr?dinger International Institute for Mathematics and Physics in Wien, given in occasion of the Workshop on New Trends in the Variational Modeling and Simulation of Liquid Crystals held at ESI, in Wien, on December 2-6, 2019. The work of the third author was supported by the EPSRC grant EP/K02390X/1 and the Leverhulme grant RPG-2018-438.

Publisher Copyright:
© 2020 Elsevier Inc.

Keywords

  • Banach function spaces
  • Grand Lebesgue spaces
  • Lebesgue spaces
  • Norm blow-up
  • Orlicz-Zygmund spaces
  • Δ condition

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