TY - JOUR

T1 - Banach algebras of Fourier multipliers equivalent at infinity to nice Fourier multipliers

AU - Fernandes, Cláudio A.

AU - Karlovich, Alexei Yu

AU - Karlovich, Yuri I.

N1 - This work was partially supported by the Fundacao para a Ciencia e a Tecnologia (Portuguese Foundation for Science and Technology) through the project UIDB/MAT/00297/2020 (Centro de Matematica e Aplicacoes) and by the SEP-CONACYT project A1-S-8793 (Mexico).

PY - 2021/4

Y1 - 2021/4

N2 - 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).

AB - 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).

KW - Banach algebra

KW - C-algebra

KW - Equivalence at infinity

KW - Fourier convolution operator

KW - Fourier multiplier

KW - Slowly oscillating function

UR - http://www.scopus.com/inward/record.url?scp=85099592697&partnerID=8YFLogxK

U2 - 10.1007/s43037-020-00111-9

DO - 10.1007/s43037-020-00111-9

M3 - Article

AN - SCOPUS:85099592697

SN - 1735-8787

VL - 15

JO - Banach Journal Of Mathematical Analysis

JF - Banach Journal Of Mathematical Analysis

IS - 2

M1 - 29

ER -