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Almost Everywhere Convergent Fourier Series

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Abstract

We study some properties of the logconvex quasi-Banach space QA defined by Arias-de-Reyna and show several applications to convergence of Fourier series. In particular, we describe the Banach envelope of QA and prove that there exists a Lorentz space strictly bigger than the Antonov space in which the almost everywhere convergence of the Fourier series holds. We also give a necessary condition for a Banach rearrangement invariant space X to be contained in QA. As an application, we show that for some classes of Banach spaces there is no the largest Banach space in a given class which is contained in QA.

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Authors and Affiliations

  1. Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, 08071, Barcelona, Spain

    M. J. Carro

  2. Faculty of Mathematics and Computer Science, Adam Mickiewicz University Poznań, and Institute of Mathematics Polish Academy of Sciences (Poznań branch), Umultowska 87, 61-614, Poznań, Poland

    M. Mastyło

  3. Faculty of Mathematics, University of Sevilla, Aptdo. de correos 1160, 41080, Sevilla, Spain

    L. Rodríguez-Piazza

Authors
  1. M. J. Carro
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  2. M. Mastyło
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  3. L. Rodríguez-Piazza
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Corresponding author

Correspondence to M. J. Carro.

Additional information

Communicated by Tom Koerner.

To the memory of Nigel Kalton.

The first author was supported by MTM2010-14946, the second author by Committee of Scientific-Research, Poland, grant No. 201 385034 and the third author by MTM2009-08934.

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Carro, M.J., Mastyło, M. & Rodríguez-Piazza, L. Almost Everywhere Convergent Fourier Series. J Fourier Anal Appl 18, 266–286 (2012). https://doi.org/10.1007/s00041-011-9199-9

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  • DOI: https://doi.org/10.1007/s00041-011-9199-9

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