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 $\mathcal {M}_{X(\mathbb {R})}$.