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