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

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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.

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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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Authors and Affiliations

  1. Department of Mathematics, University of Sussex, Pevensey 2, North-South Road, Brighton, BN1 9QH, UK

    David Edmunds

  2. Institute of Mathematics of the Academy of Sciences of the Czech Republic, Źitna 25, 115 67, Prague 1, Czech Republic

    Amiran Gogatishvili

  3. Faculty of Exact and Natural Sciences, I. Javakhishvili Tbilisi State University, University St. 2, 0143, Tbilisi, Georgia

    Tengiz Kopaliani

Authors
  1. David Edmunds
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  2. Amiran Gogatishvili
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  3. Tengiz Kopaliani
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Correspondence to Amiran Gogatishvili.

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Communicated by Mieczysław Mastyło.

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Edmunds, D., Gogatishvili, A. & Kopaliani, T. Construction of Function Spaces Close to \(L^\infty \) with Associate Space Close to \(L^1\). J Fourier Anal Appl 24, 1539–1553 (2018). https://doi.org/10.1007/s00041-017-9574-2

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  • DOI: https://doi.org/10.1007/s00041-017-9574-2

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