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Journal of Fourier Analysis and Applications

, Volume 24, Issue 6, pp 1539–1553 | Cite as

Construction of Function Spaces Close to \(L^\infty \) with Associate Space Close to \(L^1\)

  • David Edmunds
  • Amiran Gogatishvili
  • Tengiz Kopaliani
Article
  • 146 Downloads

Abstract

The paper introduces a variable exponent space X which has in common with \(L^{\infty }([0,1])\) the property that the space C([0, 1]) of continuous functions on [0, 1] is a closed linear subspace in it. The associate space of X contains both the Kolmogorov and the Marcinkiewicz examples of functions in \(L^{1}\) with a.e. divergent Fourier series.

Keywords

Banach function space Variable Lebesgue spaces a.e. divergent Fourier series Hardy–Littlewood maximal function 

Mathematics Subject Classification

46E30 42A20 

Notes

Acknowledgements

We thank the anonymous referee for his/her remarks, which have improved the final version of this paper. The research was in part supported by the Shota Rustaveli National Science Foundation (SRNSF), Grant No: 217282, Operators of Fourier analysis in some classical and new function spaces. The research of A.Gogatishvili was partially supported by the Grant P201/13/14743S of the Grant agency of the Czech Republic and RVO: 67985840.

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • David Edmunds
    • 1
  • Amiran Gogatishvili
    • 2
  • Tengiz Kopaliani
    • 3
  1. 1.Department of MathematicsUniversity of SussexBrightonUK
  2. 2.Institute of Mathematics of the Academy of Sciences of the Czech RepublicPrague 1Czech Republic
  3. 3.Faculty of Exact and Natural SciencesI. Javakhishvili Tbilisi State UniversityTbilisiGeorgia

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