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The UMD Property for Musielak–Orlicz Spaces

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Positivity and Noncommutative Analysis

Part of the book series: Trends in Mathematics ((TM))

Abstract

In this paper we show that Musielak–Orlicz spaces are UMD spaces under the so-called Δ2 condition on the generalized Young function and its complemented function. We also prove that if the measure space is divisible, then a Musielak–Orlicz space has the UMD property if and only if it is reflexive. As a consequence we show that reflexive variable Lebesgue spaces L p(⋅) are UMD spaces.

Dedicated to Ben de Pagter on the occasion of his 65th birthday

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Acknowledgements

The authors would like to thank Emiel Lorist and Jan van Neerven for helpful comments. The authors “Nick Lindemulder” and “Mark Veraar” were supported by the Vidi subsidy 639.032.427 of the Netherlands Organisation for Scientific Research (NWO).

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Authors and Affiliations

  1. Delft Institute of Applied Mathematics, Delft University of Technology, Delft, The Netherlands

    Nick Lindemulder, Mark Veraar & Ivan Yaroslavtsev

Authors
  1. Nick Lindemulder
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  2. Mark Veraar
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  3. Ivan Yaroslavtsev
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Corresponding author

Correspondence to Mark Veraar .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Mississippi, Oxford, MS, USA

    Gerard Buskes

  2. Mathematical Institute, Leiden University, Leiden, The Netherlands; Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, South Africa

    Marcel de Jeu

  3. College of Science and Engineering, Flinders University, Adelaide, Australia

    Peter Dodds

  4. Department of Mathematics, University of South Carolina, Columbia, SC, USA

    Anton Schep

  5. School of Mathematics and Statistics, University of New South Wales, Sydney, Australia

    Fedor Sukochev

  6. Delft Institute of Applied Mathematics, Delft University of Technology, Delft, The Netherlands

    Jan van Neerven

  7. Mathematical Sciences Research Centre, Queen’s University Belfast, Belfast, UK

    Anthony Wickstead

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Lindemulder, N., Veraar, M., Yaroslavtsev, I. (2019). The UMD Property for Musielak–Orlicz Spaces. In: Buskes, G., et al. Positivity and Noncommutative Analysis. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-10850-2_19

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