Algebras of Continuous Fourier Multipliers on Variable Lebesgue Spaces

Abstract

We show that several definitions of algebras of continuous Fourier multipliers on variable Lebesgue spaces over the real line are equivalent under some natural assumptions on variable exponents. Some of our results are new even in the case of standard Lebesgue spaces and give answers on two questions about algebras of continuous Fourier multipliers on Lebesgue spaces over the real line posed by Mascarenhas, Santos and Seidel.

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Acknowledgements

The author would like to thank Eugene Shargorodsky (King’s College London, UK) for interesting discussions on the subject of the paper and the anonymous referee for several useful remarks.

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Correspondence to Alexei Yu. Karlovich.

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This work was partially supported by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the project UID/MAT/00297/2019 (Centro de Matemática e Aplicações)

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Karlovich, A.Y. Algebras of Continuous Fourier Multipliers on Variable Lebesgue Spaces. Mediterr. J. Math. 17, 102 (2020). https://doi.org/10.1007/s00009-020-01537-z

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Keywords

  • Continuous Fourier multiplier
  • variable Lebesgue space
  • Stechkin’s inequality
  • piecewise continuous function
  • slowly oscillating function

Mathematics Subject Classification

  • 42B15
  • 46E30