Banach algebras of Fourier multipliers equivalent at infinity to nice Fourier multipliers

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Abstract

Let MX(R) be the Banach algebra of all Fourier multipliers on a Banach function space X(R) such that the Hardy–Littlewood maximal operator is bounded on X(R) and on its associate space X(R). For two sets Ψ, Ω⊂ MX(R), let ΨΩ be the set of those c∈ Ψ for which there exists d∈ Ω such that the multiplier norm of χR\[-N,N](c- d) tends to zero as N→ ∞. In this case, we say that the Fourier multiplier c is equivalent at infinity to the Fourier multiplier d. We show that if Ω is a unital Banach subalgebra of MX(R) consisting of nice Fourier multipliers (for instance, continuous or slowly oscillating in certain sense) and Ψ is an arbitrary unital Banach subalgebra of MX(R), then ΨΩ is a also a unital Banach subalgebra of MX(R).

Original languageEnglish
Article number29
JournalBanach Journal Of Mathematical Analysis
Volume15
Issue number2
DOIs
Publication statusPublished - Apr 2021

Keywords

  • Banach algebra
  • C-algebra
  • Equivalence at infinity
  • Fourier convolution operator
  • Fourier multiplier
  • Slowly oscillating function

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