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Archiv der Mathematik

, Volume 112, Issue 1, pp 93–100 | Cite as

Complementation of the subspace of radial multipliers in the space of Fourier multipliers on \(\mathbb {R}^n\)

  • Cédric Arhancet
  • Christoph KrieglerEmail author
Article
  • 21 Downloads

Abstract

In this short note, we prove that the subspace of radial multipliers is contractively complemented in the space of Fourier multipliers on the Bochner space \(\mathrm {L}^p(\mathbb {R}^n,X)\) where X is a Banach space and where \(1 \le p <\infty \). Moreover, if \(X = \mathbb {C}\), then this complementation preserves the positivity of multipliers.

Keywords

\(\mathrm {L}^p\)-spaces Fourier multipliers Complemented subspaces Radial multipliers 

Mathematics Subject Classification

Primary 42B15 Secondary 43A15 43A22 

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References

  1. 1.
    Bourbaki, N.: Integration. I. Chapters 1–6. Translated from the 1959, 1965 and 1967 French originals by Sterling K. Berberian. Elements of Mathematics. Springer, Berlin (2004)Google Scholar
  2. 2.
    Bourbaki, N.: Integration. II. Chapters 7–9. Translated from the 1963 and 1969 French originals by Sterling K. Berberian. Elements of Mathematics. Springer, Berlin (2004)Google Scholar
  3. 3.
    Edwards, R.E.: Functional Analysis. Theory and Applications. Corrected Reprint of the 1965 Original. Dover Publications, Inc., New York (1995)Google Scholar
  4. 4.
    Eisner, T., Farkas, B., Haase, M., Nagel, R.: Operator Theoretic Aspects of Ergodic Theory. Graduate Texts in Mathematics, vol. 272. Springer, Cham (2015)zbMATHGoogle Scholar
  5. 5.
    Hao, C.: Lecture Notes on Introduction to Harmonic Analysis. Institute of Mathematics, AMSS, CAS (2012)Google Scholar
  6. 6.
    Hytönen, T., van Neerven, J., Veraar, M., Weis, L.: Analysis on Banach Spaces. Volume I: Martingales and Littlewood-Paley Theory. Springer, Cham (2016)CrossRefzbMATHGoogle Scholar
  7. 7.
    Megginson, R.E.: An Introduction to Banach Space Theory. Graduate Texts in Mathematics, vol. 183. Springer, New York (1998)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.AlbiFrance
  2. 2.Laboratoire de Mathématiques Blaise Pascal (UMR 6620)Université Clermont AuvergneClermont-FerrandFrance

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