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

Abstract

Let \(\mathcal {M}_{X(\mathbb {R})}\) be the Banach algebra of all Fourier multipliers on a Banach function space \(X(\mathbb {R})\) such that the Hardy–Littlewood maximal operator is bounded on \(X(\mathbb {R})\) and on its associate space \(X'(\mathbb {R})\). For two sets \(\varPsi ,\varOmega \subset \mathcal {M}_{X(\mathbb {R})}\), let \(\varPsi _\varOmega\) be the set of those \(c\in \varPsi\) for which there exists \(d\in \varOmega\) such that the multiplier norm of \(\chi _{\mathbb {R}\setminus [-N,N]}(c-d)\) tends to zero as \(N\rightarrow \infty\). In this case, we say that the Fourier multiplier c is equivalent at infinity to the Fourier multiplier d. We show that if \(\varOmega\) is a unital Banach subalgebra of \(\mathcal {M}_{X(\mathbb {R})}\) consisting of nice Fourier multipliers (for instance, continuous or slowly oscillating in certain sense) and \(\varPsi\) is an arbitrary unital Banach subalgebra of \(\mathcal {M}_{X(\mathbb {R})}\), then \(\varPsi _\varOmega\) is a also a unital Banach subalgebra of \(\mathcal {M}_{X(\mathbb {R})}\).

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Acknowledgements

We would like to thank an anonymous referee who suggested us a stronger form of the main result and a simplification of its proof.

Funding

This work was partially supported by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the project UIDB/MAT/ 00297/2020 (Centro de Matemática e Aplicações) and by the SEP-CONACYT project A1-S-8793 (México).

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Correspondence to Alexei Yu. Karlovich.

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Communicated by Luis Castro.

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Fernandes, C.A., Karlovich, A.Y. & Karlovich, Y.I. Banach algebras of Fourier multipliers equivalent at infinity to nice Fourier multipliers. Banach J. Math. Anal. 15, 29 (2021). https://doi.org/10.1007/s43037-020-00111-9

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Keywords

  • Fourier convolution operator
  • Fourier multiplier
  • Slowly oscillating function
  • Equivalence at infinity
  • Banach algebra
  • \(C^*\)-algebra

Mathematics Subject Classification

  • 47G10
  • 42A45
  • 46E30