Semi-almost periodic Fourier multipliers on rearrangement-invariant spaces with suitable Muckenhoupt weights

Abstract

Let $X({\mathbb {R}})$ be a separable rearrangement-invariant space and w be a suitable Muckenhoupt weight. We show that for any semi-almost periodic Fourier multiplier a on $X({\mathbb {R}},w)=\{f:fw\in X({\mathbb {R}})\}$ there exist uniquely determined almost periodic Fourier multipliers $a_l,a_r$ on $X({\mathbb {R}},w)$, such that

$$\begin{aligned} a=(1-u)a_l+ua_r+a_0, \end{aligned}$$

for some monotonically increasing function u with $u(-\infty )=0$, $u(+\infty )=1$ and some continuous and vanishing at infinity Fourier multiplier $a_0$ on $X({\mathbb {R}},w)$. This result extends previous results by Sarason (Duke Math J 44:357–364, 1977) for $L^2({\mathbb {R}})$ and by Karlovich and Loreto Hernández (Integral Equ Oper Theor 62:85–128, 2008) for weighted Lebesgue spaces $L^p({\mathbb {R}},w)$ with weights in a suitable subclass of the Muckenhoupt class $A_p({\mathbb {R}})$.

1 Introduction

Let $C({\overline{{\mathbb {R}}}})$ be the $C^*$-algebra of all continuous functions on the two-point compactification of the real line ${\overline{{\mathbb {R}}}}=[-\infty ,+\infty ]$ and

$$\begin{aligned} C({{\dot{{\mathbb {R}}}}})=\{f\in C({\overline{{\mathbb {R}}}}):f(-\infty )=f(+\infty )\}, \end{aligned}$$

where ${{\dot{{\mathbb {R}}}}}={\mathbb {R}}\cup \{\infty \}$ is the one-point compactification of the real line. Let APP denote the set of all almost periodic polynomials, that is, finite sums of the form $\sum _{\lambda \in \varLambda }c_\lambda e_\lambda$, where

$$\begin{aligned} e_\lambda (x):=e^{i\lambda x}, \quad x\in {\mathbb {R}}, \end{aligned}$$

$c_\lambda \in {\mathbb {C}}$ and $\varLambda \subset {\mathbb {R}}$ is a finite subset of ${\mathbb {R}}$. The smallest closed subalgebra of $L^\infty ({\mathbb {R}})$ that contains APP is denoted by AP and called the algebra of (uniformly) almost periodic functions. Sarason [36] introduced the algebra of semi-almost periodic functions as the smallest closed subalgebra of $L^\infty ({\mathbb {R}})$ that contains AP and $C({\overline{{\mathbb {R}}}})$:

$$\begin{aligned} SAP:={\text {alg}}_{L^\infty ({\mathbb {R}})}\{AP,C({\overline{{\mathbb {R}}}})\}. \end{aligned}$$

It is not difficult to see that AP and SAP are $C^*$-subalgebras of $L^\infty ({\mathbb {R}})$.

Theorem 1.1

(Sarason [36], see also [10, Theorem 1.21]) Let $u\in C({\overline{{\mathbb {R}}}})$ be any function for which $u(-\infty )=0$ and $u(+\infty )=1$. If $a\in SAP$, then there exist $a_l,a_r\in AP$ and $a_0\in C({{\dot{{\mathbb {R}}}}})$ such that $a_0(\infty )=0$ and

$$\begin{aligned} a=(1-u)a_l+ua_r+a_0. \end{aligned}$$

(1.1)

The functions $a_l,a_r$ are uniquely determined by a and independent of the particular choice of u. The maps $a\mapsto a_l$ and $a\mapsto a_r$ are $C^*$-algebra homomorphisms of SAP onto AP.

The uniquely determined function $a_l$ (resp. $a_r$) is called the left (resp. right) almost periodic representative of the semi-almost periodic function a.

Let $\mathcal{F}:L^2({\mathbb {R}})\rightarrow L^2({\mathbb {R}})$ denote the Fourier transform:

$$\begin{aligned} ( \mathcal{F}f)(x):={\widehat{f}}(x):=\int _{\mathbb {R}}f(t)e^{itx}\,\mathrm{d}t, \quad x\in {\mathbb {R}}, \end{aligned}$$

and let $\mathcal{F}^{-1}:L^2({\mathbb {R}})\rightarrow L^2({\mathbb {R}})$ be the inverse of $\mathcal{F}$,

$$\begin{aligned} ( \mathcal{F}^{-1}g)(t)=\frac{1}{2\pi }\int _{\mathbb {R}}g(x)e^{-itx}\,\mathrm{d} x, \quad t\in {\mathbb {R}}. \end{aligned}$$

It is well known that the Fourier convolution operator

$$\begin{aligned} W^0(a):=\mathcal{F}^{-1}a\mathcal{F} \end{aligned}$$

(1.2)

is bounded on the space $L^2({\mathbb {R}})$ for every $a\in L^\infty ({\mathbb {R}})$.

Let $X({\mathbb {R}})$ be a separable Banach function space (see Sect. 2.1 for the definition and some properties of Banach function spaces). Then $L^2({\mathbb {R}})\cap X({\mathbb {R}})$ is dense in $X({\mathbb {R}})$ (see, e.g., [15, Lemma 2.1]). A function $a\in L^\infty ({\mathbb {R}})$ is called a Fourier multiplier on $X({\mathbb {R}})$ if the convolution operator $W^0(a)$ defined by (1.2) maps the set $L^2({\mathbb {R}})\cap X({\mathbb {R}})$ into the space $X({\mathbb {R}})$ and extends to a bounded linear operator on $X({\mathbb {R}})$. The function a is called the symbol of the Fourier convolution operator $W^0(a)$. The set ${\mathcal {M}}_{X({\mathbb {R}})}$ of all Fourier multipliers on $X({\mathbb {R}})$ is a unital normed algebra under pointwise operations and the norm:

$$\begin{aligned} \left\| a\right\| _{{\mathcal {M}}_{X({\mathbb {R}})}}:=\left\| W^0(a)\right\| _{{\mathcal {B}}(X({\mathbb {R}}))}, \end{aligned}$$

where ${\mathcal {B}}(X({\mathbb {R}}))$ denotes the Banach algebra of all bounded linear operators on the space $X({\mathbb {R}})$.

Note that the Lebesgue spaces $L^p({\mathbb {R}})$, $1\le p\le \infty$, constitute the simplest example of Banach function spaces. Motivated by the work of Duduchava and Saginashvili [14], Karlovich and Spitkovsky [29] (see also [10, Section 19.1]) introduced the algebra $SAP_{L^p({\mathbb {R}})}$ of semi-almost periodic Fourier multipliers on the Lebesgue spaces $L^p({\mathbb {R}})$, $1<p<\infty$, and proved an analogue of Sarason’s Theorem 1.1 for $SAP_{L^p({\mathbb {R}})}$ (see [29, Lemma 3.1(iv)] and [10, Proposition 19.3]).

We should mention that, after Sarason’s pioneering paper [36], various classes of Toeplitz and convolution type operators involving semi-almost periodic functions were studied on various function spaces, for instance, by Saginashvili [35], Grudsky [19]; Böttcher et al. [3,4,5,6, 8,9,10]; Nolasco and Castro [32, 33]; Bogveradze and Castro [2]; the second author and Spitkovsky [25].

Let ${\mathfrak {M}}({\mathbb {R}})$ denote the set of all measurable complex-valued Lebesgue measurable functions on ${\mathbb {R}}$. As usual, we identify two functions on ${\mathbb {R}}$ which are equal almost everywhere. A measurable function $w:{\mathbb {R}}\rightarrow [0,\infty ]$ is called a weight if the set $w^{-1}(\{0,\infty \})$ has measure zero. For $1<p<\infty$, the Muckenhoupt class $A_p({\mathbb {R}})$ is defined as the class of all weights $w:{\mathbb {R}}\rightarrow [0,\infty ]$ such that $w\in L_{\mathrm{loc}}^p({\mathbb {R}})$, $w^{-1}\in L_{\mathrm{loc}}^{p'}({\mathbb {R}})$ and

$$\begin{aligned} \sup _I \left( \frac{1}{|I|}\int _I w^p(x)\,\mathrm{d}x\right) ^{1/p} \left( \frac{1}{|I|}\int _I w^{-p'}(x)\,\mathrm{d}x\right) ^{1/p'} <\infty , \end{aligned}$$

(1.3)

where $1/p+1/p'=1$ and the supremum is taken over all intervals $I\subset {\mathbb {R}}$ of finite length |I|. Since $w\in L_{\mathrm{loc}}^p({\mathbb {R}})$ and $w^{-1}\in L_{\mathrm{loc}}^{p'}({\mathbb {R}})$, the weighted Lebesgue space

$$\begin{aligned} L^p({\mathbb {R}},w):=\{f\in {\mathfrak {M}}({\mathbb {R}}):fw\in L^p({\mathbb {R}})\} \end{aligned}$$

is a separable Banach function space (see, e.g., [26, Lemma 2.4]) with the norm:

$$\begin{aligned} \Vert f\Vert _{L^p({\mathbb {R}},w)}:=\left( \int _{\mathbb {R}}|f(x)|^p w^p(x)\,\mathrm{d}x\right) ^{1/p}. \end{aligned}$$

Note that if $w\in A_p({\mathbb {R}})$, then it may happen that the function $e_\lambda$ does not belong to ${\mathcal {M}}_{L^p({\mathbb {R}},w)}$ for some $\lambda \in {\mathbb {R}}$. Hence,order to generalize Theorem 1.1 to the setting of weighted Lebesgue spaces $L^p({\mathbb {R}},w)$, one has to restrict the study to a narrower class of weights. For $1<p<\infty$, let

$$\begin{aligned} A_p^0({\mathbb {R}}):=\left\{ w\in A_p({\mathbb {R}})\ :\ v_\lambda =\frac{w(\cdot +\lambda )}{w(\cdot )}\in L^\infty ({\mathbb {R}}) \text{ for } \text{ all } \lambda \in {\mathbb {R}}\right\} . \end{aligned}$$

For a weight $w\in A_p^0({\mathbb {R}})$, Karlovich and Loreto Hernández defined the algebra $SAP_{L^p({\mathbb {R}},w)}$ of semi-almost periodic Fourier multipliers on the weighted Lebesgue space $L^p({\mathbb {R}},w)$ and proved an analogue of Theorem 1.1 in this setting (see [27, Theorem 3.1]). The aim of this paper is to extend this result to the setting of separable rearrangement-invariant Banach function spaces with suitable Muckenhoupt weights.

It is well known that the Lebesgue spaces $L^p({\mathbb {R}})$, $1\le p\le \infty$, fall in the class of rearrangement-invariant Banach function spaces. Other classical examples of rearrangement-invariant Banach function spaces are Orlicz spaces $L^\varPhi ({\mathbb {R}})$ and Lorentz spaces $L^{p,q}({\mathbb {R}})$, $1\le p,q\le \infty$. For a rearrangement-invariant Banach function space $X({\mathbb {R}})$, its Boyd indices $\alpha _X,\beta _X$ are important interpolation characteristics. In particular, $\alpha _{L^p}=\beta _{L^p}=1/p$ for $1\le p\le \infty$. In general, $0\le \alpha _X\le \beta _X\le 1$ and it may happen that $\alpha _X<\beta _X$. We postpone formal definitions of rearrangement-invariant Banach function spaces and their Boyd indices until Sects. 2.2–2.3 and refer to [1, Chap. 3] and [30, Chap. 2] for the detailed study of these concepts.

Let $X({\mathbb {R}})$ be a separable rearrangement-invariant Banach function space with the Boyd indices $\alpha _X,\beta _X$ satisfying $0<\alpha _X,\beta _X<1$. Suppose that a weight w belongs to $A_{1/\alpha _X}({\mathbb {R}})\cap A_{1/\beta _X}({\mathbb {R}})$. Then

$$\begin{aligned} X({\mathbb {R}},w):=\{f\in {\mathfrak {M}}({\mathbb {R}}):fw\in X({\mathbb {R}})\} \end{aligned}$$

is a separable Banach function space (see Lemma 2.3(b) below). Suppose that $a:{\mathbb {R}}\rightarrow {\mathbb {C}}$ is a function of finite total variation V(a) given by

$$\begin{aligned} V(a):=\sup \sum _{k=1}^n |a(x_k)-a(x_{k-1})|, \end{aligned}$$

where the supremum is taken over all partitions of ${\mathbb {R}}$ of the form

$$\begin{aligned} -\infty<x_0<x_1<\cdots<x_n<+\infty \end{aligned}$$

with $n\in {\mathbb {N}}$. The set $V({\mathbb {R}})$ of all functions of finite total variation on ${\mathbb {R}}$ with the norm

$$\begin{aligned} \Vert a\Vert _V:=\Vert a\Vert _{L^\infty ({\mathbb {R}})}+V(a) \end{aligned}$$

is a unital non-separable Banach algebra. It follows from [21, Corollary 2.2] that there exists a constant $c_{X({\mathbb {R}},w)}\in (0,\infty )$ such that for all $a\in V({\mathbb {R}})$,

$$\begin{aligned} \Vert a\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}\le c_{X({\mathbb {R}},w)}\Vert a\Vert _{V({\mathbb {R}})}. \end{aligned}$$

(1.4)

This inequality is usually called a Stechkin-type inequality (see, e.g., [13, Theorem 2.11] and [10, Theorem 17.1] for the case of Lebesgue spaces and Lebesgue spaces with Muckenhoupt weights, respectively). Let $C_{X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}})$ and $C_{X({\mathbb {R}},w)}({\overline{{\mathbb {R}}}})$ denote the closures of $C({{\dot{{\mathbb {R}}}}})\cap V({\mathbb {R}})$ and $C({\overline{{\mathbb {R}}}})\cap V({\mathbb {R}})$ with respect to the norm of ${\mathcal {M}}_{X({\mathbb {R}},w)}$, respectively.

If $w\in A_{1/\alpha _X}^0({\mathbb {R}})\cap A_{1/\beta _X}^0({\mathbb {R}})$, then $APP\subset {\mathcal {M}}_{X({\mathbb {R}},w)}$ (see Corollary 5.2 below). Because of this observation, we will refer to $A_{1/\alpha _X}^0({\mathbb {R}})\cap A_{1/\beta _X}^0({\mathbb {R}})$ as the class of suitable Muckenhoupt weights. By $AP_{X({\mathbb {R}},w)}$ we denote the closure of APP with respect to the norm of ${\mathcal {M}}_{X({\mathbb {R}},w)}$. Finally, let $SAP_{X({\mathbb {R}},w)}$ be the smallest closed subalgbera of ${\mathcal {M}}_{X({\mathbb {R}},w)}$ that contains the algebras $AP_{X({\mathbb {R}},w)}$ and $C_{X({\mathbb {R}},w)}({\overline{{\mathbb {R}}}})$:

$$\begin{aligned} SAP_{X({\mathbb {R}},w)}={\text {alg}}_{{\mathcal {M}}_{X({\mathbb {R}},w)}} \left\{ AP_{X({\mathbb {R}},w)},C_{X({\mathbb {R}},w)}({\overline{{\mathbb {R}}}})\right\} . \end{aligned}$$

In this paper we present a self-contained proof of the following result.

Theorem 1.2

(Main result) Let $X({\mathbb {R}})$ be a separable rearrangement-invariant Banach function space with the Boyd indices satisfying $0<\alpha _X$, $\beta _X<1$. Suppose that $w\in A_{1/\alpha _X}^0({\mathbb {R}})\cap A_{1/\beta _X}^0({\mathbb {R}})$. Let $u\in C({\overline{{\mathbb {R}}}})$ be any real-valued monotonically increasing function such that $u(-\infty )=0$ and $u(+\infty )=1$. Then for every function $a\in SAP_{X({\mathbb {R}},w)}$ there exist functions $a_l,a_r\in AP_{X({\mathbb {R}},w)}$ and a function $a_0\in C_{X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}})$ such that $a_0(\infty )=0$ and (1.1) holds. The functions $a_l$, $a_r$ are uniquely determined by the function a and are independent of the particular choice of the function u. The maps $a\mapsto a_l$ and $a\mapsto a_r$ are continuous Banach algebra homomorphisms of $SAP_{X({\mathbb {R}},w)}$ onto $AP_{X({\mathbb {R}},w)}$ of norm 1.

The paper is organized as follows. In Sect. 2, we collect definitions and properties of rearrangement-invariant Banach functions spaces and their Boyd indices $\alpha _X,\beta _X$. Further, we discuss properties of weighted rearrangement-invariant spaces $X({\mathbb {R}},w)$ and state several results about general Fourier multipliers on $X({\mathbb {R}},w)$ for weights w belonging to the intersection of the Muckenhoupt classes $A_{1/\alpha _X}({\mathbb {R}})\cap A_{1/\beta _X}({\mathbb {R}})$.

In Sect. 3, we show that, under the assumption $w\in A_{1/\alpha _X}({\mathbb {R}})\cap A_{1/\beta _X}({\mathbb {R}})$, the set of continuous Fourier multipliers vanishing at infinity on the space $X({\mathbb {R}},w)$ coincides with the closure of the set of all smooth compactly supported functions with respect to the norm of ${\mathcal {M}}_{X({\mathbb {R}},w)}$.

Relying on the results of the previous section, in Sect. 4, we show that $C_{X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}})=C_{X({\mathbb {R}},w)}({\overline{{\mathbb {R}}}})\cap C({{\dot{{\mathbb {R}}}}})$ and that the algebra $C_{X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}})$ is contained in the algebra $SO_{X({\mathbb {R}},w)}$ of slowly oscillating Fourier multipliers (see [21]).

In Sect. 5, we show that if $w\in A_{1/\alpha _X}^0({\mathbb {R}})\cap A_{1/\beta _X}^0({\mathbb {R}})$, then the set of almost periodic polynomials APP is contained in ${\mathcal {M}}_{X({\mathbb {R}},w)}$. We give an example of a nontrivial weight in $A_{1/\alpha _X}^0({\mathbb {R}})\cap A_{1/\beta _X}^0({\mathbb {R}})$ (based on an example from [27]). Further, we show that the product of an almost periodic Fourier multiplier and a continuous Fourier multiplier vanishing at infinity is a continuous Fourier multiplier vanishing at infinity.

Section 6 is devoted to the proof of the main result. We show that the set ${\mathcal {A}}_u$ of functions of the form (1.1) with $a_l,a_r\in AP_{X({\mathbb {R}},w)}$ and $a_0\in C_{X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}})$ such that $a_0(\infty )=0$ forms an algebra, and that the mappings $a\mapsto a_l$ and $a\mapsto a_r$ are algebraic homomorphisms of ${\mathcal {A}}_u$ onto $AP_{X({\mathbb {R}},w)}$. We prove that

$$\begin{aligned} \Vert a_l\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}\le \Vert a\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}, \quad \Vert a_r\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}\le \Vert a\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}, \quad a\in {\mathcal {A}}_u, \end{aligned}$$

(1.5)

which implies that the algebra ${\mathcal {A}}_u$ is closed. Since the closure of ${\mathcal {A}}_u$ with respect to the norm of ${\mathcal {M}}_{X({\mathbb {R}},w)}$ coincides with $SAP_{X({\mathbb {R}},w)}$, we conclude that ${\mathcal {A}}_u$ is equal to $SAP_{X({\mathbb {R}},w)}$. Moreover, inequalities (1.5) mean that $a\mapsto a_l$ and $a\mapsto a_r$ are Banach algebra homomorphisms of $SAP_{X(|R,w)}$ onto $AP_{X({\mathbb {R}},w)}$ of norm 1.

2 Preliminaries

2.1 Banach function spaces

Let ${\mathbb {R}}_+:=(0,\infty )$ and ${\mathbb {S}}\in \{{\mathbb {R}}_+,{\mathbb {R}}\}$. The set of all Lebesgue measurable complex-valued functions on ${\mathbb {S}}$ is denoted by ${\mathfrak {M}}({\mathbb {S}})$. Let ${\mathfrak {M}}^+({\mathbb {S}})$ be the subset of functions in ${\mathfrak {M}}({\mathbb {S}})$ whose values lie in $[0,\infty ]$. The Lebesgue measure of a measurable set $E\subset {\mathbb {S}}$ is denoted by |E| and its characteristic function is denoted by $\chi _E$. Following [1, Chap. 1, Definition 1.1], a mapping $\rho :{\mathfrak {M}}^+({\mathbb {S}})\rightarrow [0,\infty ]$ is called a Banach function norm if, for all functions $f,g, f_n \ (n\in {\mathbb {N}})$ in ${\mathfrak {M}}^+({\mathbb {S}})$, for all constants $a\ge 0$, and for all measurable subsets E of ${\mathbb {S}}$, the following properties hold:

$$\begin{aligned} \mathrm{(A1)}\;&\rho (f)=0 \Leftrightarrow f=0\ \text{ a.e.}, \quad \rho (af)=a\rho (f), \quad \rho (f+g) \le \rho (f)+\rho (g),\\ \mathrm{(A2)}\;&0\le g \le f \ \text{ a.e. } \ \Rightarrow \ \rho (g) \le \rho (f) \quad \text{(the } \text{ lattice } \text{ property)}, \\ \mathrm{(A3)}\;&0\le f_n \uparrow f \ \text{ a.e. } \ \Rightarrow \ \rho (f_n) \uparrow \rho (f)\quad \text{(the } \text{ Fatou } \text{ property)},\\ \mathrm{(A4)}\;&|E|<\infty \Rightarrow \rho (\chi _E)<\infty ,\\ \mathrm{(A5)}\;&|E|<\infty \Rightarrow \int _E f(x)\,\mathrm{d}x \le C_E\rho (f) \end{aligned}$$

with $C_E \in (0,\infty )$ which may depend on E and $\rho$ but is independent of f. When functions differing only on a set of measure zero are identified, the set $X({\mathbb {S}})$ of all functions $f\in {\mathfrak {M}}({\mathbb {S}})$ for which $\rho (|f|)<\infty$ is called a Banach function space. For each $f\in X({\mathbb {S}})$, the norm of f is defined by

$$\begin{aligned} \left\| f\right\| _{X({\mathbb {S}})} :=\rho (|f|). \end{aligned}$$

Under the natural linear space operations and under this norm, the set $X({\mathbb {S}})$ becomes a Banach space (see [1, Chap. 1, Theorems 1.4 and 1.6]). If $\rho$ is a Banach function norm, its associate norm $\rho '$ is defined on ${\mathfrak {M}}^+({\mathbb {S}})$ by

$$\begin{aligned} \rho '(g):=\sup \left\{ \int _{{\mathbb {S}}} f(x)g(x)\,dx \ : \ f\in {\mathfrak {M}}^+({\mathbb {S}}), \ \rho (f) \le 1 \right\} , \quad g\in {\mathfrak {M}}^+({\mathbb {S}}). \end{aligned}$$

It is a Banach function norm itself [1, Chap. 1, Theorem 2.2]. The Banach function space $X'({\mathbb {R}})$ determined by the Banach function norm $\rho '$ is called the associate space (Köthe dual) of $X({\mathbb {S}})$. The associate space $X'({\mathbb {S}})$ is naturally identified with a subspace of the (Banach) dual space $[X({\mathbb {S}})]^*$.

2.2 Rearrangement-invariant Banach function spaces

Suppose that ${\mathbb {S}}\in \{{\mathbb {R}},{\mathbb {R}}_+\}$. Let ${\mathfrak {M}}_0({\mathbb {S}})$ and ${\mathfrak {M}}_0^+({\mathbb {S}})$ be the classes of a.e. finite functions in ${\mathfrak {M}}({\mathbb {S}})$ and ${\mathfrak {M}}^+({\mathbb {S}})$, respectively. The distribution function $\mu _f$ of a function $f\in {\mathfrak {M}}_0({\mathbb {S}})$ is given by

$$\begin{aligned} \mu _f(\lambda ):= \big |\{x\in {\mathbb {S}}:|f(x)|>\lambda \}\big |, \quad \lambda \ge 0. \end{aligned}$$

Two functions $f,g\in {\mathfrak {M}}_0({\mathbb {S}})$ are said to be equimeasurable if $\mu _f(\lambda )=\mu _g(\lambda )$ for all $\lambda \ge 0$. The non-increasing rearrangement of $f\in {\mathfrak {M}}_0({\mathbb {S}})$ is the function defined by

$$\begin{aligned} f^*(t):=\inf \{\lambda :\mu _f(\lambda )\le t\},\quad t\ge 0. \end{aligned}$$

We here use the standard convention that $\inf \emptyset =+\infty$.

A Banach function norm $\rho :{\mathfrak {M}}^+({\mathbb {S}}) \rightarrow [0,\infty ]$ is called rearrangement-invariant if for every pair of equimeasurable functions $f,g \in {\mathfrak {M}}^+_0({\mathbb {S}})$ the equality $\rho (f)=\rho (g)$ holds. In that case, the Banach function space $X({\mathbb {S}})$ generated by $\rho$ is said to be a rearrangement-invariant Banach function space (or simply rearrangement-invariant space). Lebesgue, Orlicz, and Lorentz spaces are classical examples of rearrangement-invariant Banach function spaces (see, e.g., [1] and the references therein). By [1, Chap. 2, Proposition 4.2], if a Banach function space $X({\mathbb {S}})$ is rearrangement-invariant, then its associate space $X'({\mathbb {S}})$ is rearrangement-invariant, too.

2.3 Boyd indices

Suppose $X({\mathbb {R}})$ is a rearrangement-invariant Banach function space generated by a rearrangement-invariant Banach function norm $\rho$. In this case, the Luxemburg representation theorem [1, Chap. 2, Theorem 4.10] provides a unique rearrangement-invariant Banach function norm ${\overline{\rho }}$ over the half-line ${\mathbb {R}}_+$ equipped with the Lebesgue measure, defined by

$$\begin{aligned} {\overline{\rho }}(h):= \sup \left\{ \int _{{\mathbb {R}}_+} g^*(t) h^*(t)\,\mathrm{d}t: \ \rho '(g)\le 1\right\} , \end{aligned}$$

and such that $\rho (f)={\overline{\rho }}(f^*)$ for all $f\in {\mathfrak {M}}_0^+({\mathbb {R}})$. The rearrangement-invariant Banach function space generated by ${\overline{\rho }}$ is denoted by ${\overline{X}}({\mathbb {R}}_+)$.

For each $t>0$, let $E_t$ denote the dilation operator defined on ${\mathfrak {M}}({\mathbb {R}}_+)$ by

$$\begin{aligned} (E_tf)(s)=f(st),\quad 0<s<\infty . \end{aligned}$$

With $X({\mathbb {R}})$ and ${\overline{X}}({\mathbb {R}}_+)$ as above, let $h_X(t)$ denote the operator norm of $E_{1/t}$ as an operator on ${\overline{X}}({\mathbb {R}}_+)$. By [1, Chap. 3, Proposition 5.11], for each $t>0$, the operator $E_t$ is bounded on ${\overline{X}}({\mathbb {R}}_+)$ and the function $h_X$ is increasing and submultiplicative on $(0,\infty )$. The Boyd indices of $X({\mathbb {R}})$ are the numbers $\alpha _X$ and $\beta _X$ defined by

$$\begin{aligned} \alpha _X:=\sup _{t\in (0,1)}\frac{\log h_X(t)}{\log t}, \quad \beta _X:=\inf _{t\in (1,\infty )}\frac{\log h_X(t)}{\log t}. \end{aligned}$$

By [1, Chap. 3, Proposition 5.13], $0\le \alpha _X\le \beta _X\le 1$. The Boyd indices are said to be nontrivial if $\alpha _X,\beta _X\in (0,1)$. The Boyd indices of the Lebesgue space $L^p({\mathbb {R}})$, $1\le p\le \infty$, are both equal to 1/p. Note that the Boyd indices of a rearrangement-invariant space may be different [1, Chap. 3, Exercises 6, 13].

The next theorem follows from the Boyd interpolation theorem [11, Theorem 1] for quasi-linear operators of weak types (p, p) and (q, q). Its proof can also be found in [1, Chap. 3, Theorem 5.16] and [30, Theorem 2.b.11].

Theorem 2.1

Let $1\le q<p\le \infty$ and $X({\mathbb {R}})$ be a rearrangement-invariant Banach function space with the Boyd indices $\alpha _X,\beta _X$ satisfying $1/p<\alpha _X$, $\beta _X<1/q$. Then there exists a constant $C_{p,q}\in (0,\infty )$ such that if a linear operator $T:{\mathfrak {M}}({\mathbb {R}})\rightarrow {\mathfrak {M}}({\mathbb {R}})$ is bounded on the Lebesgue spaces $L^p({\mathbb {R}})$ and $L^q({\mathbb {R}})$, then it is also bounded on the rearrangement-invariant Banach function space $X({\mathbb {R}})$ and

$$\begin{aligned} \Vert T\Vert _{{\mathcal {B}}(X({\mathbb {R}}))}\le C_{p,q}\max \big \{ \Vert T\Vert _{{\mathcal {B}}(L^p({\mathbb {R}}))},\Vert T\Vert _{{\mathcal {B}}(L^q({\mathbb {R}}))} \big \}. \end{aligned}$$

(2.1)

Notice that estimate (2.1) is not stated explicitly in [1, 11, 30]. However, it can be extracted from the proof of the Boyd interpolation theorem.

2.4 Weighted Banach function spaces

Let $X({\mathbb {R}})$ be a Banach function space generated by a Banach function norm $\rho$. We say that $f\in X_{\mathrm{loc}}({\mathbb {R}})$ if $f\chi _E\in X({\mathbb {R}})$ for any measurable set $E\subset {\mathbb {R}}$ of finite measure.

Lemma 2.2

[26, Lemma 2.4] Let $X({\mathbb {R}})$ be a Banach function space generated by a Banach function norm $\rho$, let $X'({\mathbb {R}})$ be its associate space, and let $w:{\mathbb {R}}\rightarrow [0,\infty ]$ be a weight. Suppose that $w\in X_{\mathrm{loc}}({\mathbb {R}})$ and $1/w\in X_{\mathrm{loc}}'({\mathbb {R}})$. Then

$$\begin{aligned} \rho _w(f):=\rho (fw),\quad f\in {\mathfrak {M}}^+({\mathbb {R}}), \end{aligned}$$

is a Banach function norm and

$$\begin{aligned} X({\mathbb {R}},w):=\{f\in {\mathfrak {M}}({\mathbb {R}}):fw\in X({\mathbb {R}})\} \end{aligned}$$

is a Banach function space generated by the Banach function norm $\rho _w$. The space $X'({\mathbb {R}},w^{-1})$ is the associate space of $X({\mathbb {R}},w)$.

2.5 Density of nice functions in separable rearrangement-invariant Banach function spaces with Muckenhoupt weights

Recall that the (noncentered) Hardy–Littlewood maximal function Mf of a function $f\in L_{\mathrm{loc}}^1({\mathbb {R}})$ is defined by

$$\begin{aligned} (Mf)(x):=\sup _{I\ni x}\frac{1}{|I|}\int _I|f(y)|\,\mathrm{d}y, \quad x\in {\mathbb {R}}, \end{aligned}$$

where the supremum is taken over all intervals $I\subset {\mathbb {R}}$ of finite length containing the point x.

Let ${\mathcal {S}}({\mathbb {R}})$ be the Schwartz space of rapidly decreasing smooth functions and let us denote by ${\mathcal {S}}_0({\mathbb {R}})$ the set of all functions $f\in {\mathcal {S}}({\mathbb {R}})$ such that their Fourier transforms ${\mathcal {F}}f$ have compact supports.

Lemma 2.3

Let $X({\mathbb {R}})$ be a separable rearrangement-invariant Banach function space and $X'({\mathbb {R}})$ be its associate space. Suppose that the Boyd indices of $X({\mathbb {R}})$ satisfy $0<\alpha _X$, $\beta _X<1$ and $w\in A_{1/\alpha _X}({\mathbb {R}})\cap A_{1/\beta _X}({\mathbb {R}})$. Then

• (a) $w\in X_{\mathrm{loc}}({\mathbb {R}})$ and $1/w\in X_{\mathrm{loc}}'({\mathbb {R}})$;

• (b) the Banach function space space $X({\mathbb {R}},w)$ is separable;

• (c) the Hardy-Littlewood maximal operator M is bounded on the Banach function space $X({\mathbb {R}},w)$ and on its associate space $X'({\mathbb {R}},w^{-1})$;

• (d) the set ${\mathcal {S}}_0({\mathbb {R}})$ is dense in the Banach function space $X({\mathbb {R}},w)$.

Proof

Parts (a) and (c) are proved in [21, Section 4.3]. Part (b) follows from part (a), Lemma 2.2 and [26, Lemmas 2.7 and 2.11]. Part (d) is a consequence of parts (b), (c) and [16, Theorem 4]. $\square$

2.6 The Banach algebra ${\mathcal {M}}_{X({\mathbb {R}},w)}$ of Fourier multipliers

The following result plays an important role in this paper.

Theorem 2.4

Let $X({\mathbb {R}})$ be a separable rearrangement-invariant Banach function space with the Boyd indices satisfying $0<\alpha _X$, $\beta _X<1$. Suppose that a weight w belongs to $A_{1/\alpha _X}({\mathbb {R}})\cap A_{1/\beta _X}({\mathbb {R}})$. If $a\in {\mathcal {M}}_{X({\mathbb {R}},w)}$, then

$$\begin{aligned} \Vert a\Vert _{L^\infty ({\mathbb {R}})}\le \Vert a\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}. \end{aligned}$$

(2.2)

The constant 1 on the right-hand side of (2.2) is best possible.

This theorem follows from Lemma 2.3(b) and [15, Theorem 2.4] (which was deduced from [24, Corollary 4.2]).

Inequality (2.2) was established earlier in [22, Theorem 1] with some constant on the right-hand side that depends on the space $X({\mathbb {R}},w)$.

Since (2.2) is available, an easy adaptation of the proof of [18, Proposition 2.5.13] leads to the following (we refer to the proof of [22, Corollary 1] for details).

Corollary 2.5

Let $X({\mathbb {R}})$ be a separable rearrangement-invariant Banach function space with the Boyd indices satisfying $0<\alpha _X$, $\beta _X<1$. Suppose that a weight w belongs to $A_{1/\alpha _X}({\mathbb {R}})\cap A_{1/\beta _X}({\mathbb {R}})$. Then the set of the Fourier multipliers ${\mathcal {M}}_{X({\mathbb {R}},w)}$ is a Banach algebra under pointwise operations and the norm $\Vert \cdot \Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}$.

As usual, we denote by $C_c^\infty ({\mathbb {R}})$ the set of all infinitely differentiable functions with compact support.

Theorem 2.6

Suppose that a non-negative even function $\varphi \in C_c^\infty ({\mathbb {R}})$ satisfies the condition

$$\begin{aligned} \int _{\mathbb {R}}\varphi (x)\,\mathrm{d}x=1 \end{aligned}$$

(2.3)

and the function $\varphi _\delta$ is defined for $\delta >0$ by

$$\begin{aligned} \varphi _\delta (x):=\delta ^{-1}\varphi (x/\delta ), \quad x\in {\mathbb {R}}. \end{aligned}$$

(2.4)

Let $X({\mathbb {R}})$ be a separable rearrangement-invariant Banach function space with the Boyd indices satisfying $0<\alpha _X$, $\beta _X<1$. Suppose that a weight w belongs to $A_{1/\alpha _X}({\mathbb {R}})\cap A_{1/\beta _X}({\mathbb {R}})$. If $a\in {\mathcal {M}}_{X({\mathbb {R}},w)}$, then for every $\delta >0$,

$$\begin{aligned} \Vert a*\varphi _\delta \Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}\le \Vert a\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}. \end{aligned}$$

(2.5)

Proof

The proof is analogous to the proof of [23, Theorem 2.6]. It follows from Lemma 2.3(c) and [26, Theorems 3.8(a) and 3.9(c)] that if the weight w belongs to $A_{1/\alpha _X}({\mathbb {R}})\cap A_{1/\beta _X}({\mathbb {R}})$, then

$$\begin{aligned} \sup _{-\infty<a<b<\infty }\frac{1}{b-a} \Vert \chi _{(a,b)}\Vert _{X({\mathbb {R}},w)}\Vert \chi _{(a,b)}\Vert _{X'({\mathbb {R}},w^{-1})}<\infty . \end{aligned}$$

Therefore, by [24, Lemma 1.3], the Banach function space $X({\mathbb {R}},w)$ satisfies the hypotheses of [24, Theorem 1.3]. It is shown in its proof (see [24, Section 4.2]) that for every $\delta >0$ and every $f\in {\mathcal {S}}({\mathbb {R}})\cap X({\mathbb {R}},w)$,

$$\begin{aligned} \Vert {\mathcal {F}}^{-1}(a*\varphi _\delta ){\mathcal {F}}f\Vert _{X({\mathbb {R}},w)} \le \sup \left\{ \frac{\Vert {\mathcal {F}}^{-1}a{\mathcal {F}}f\Vert _{X({\mathbb {R}},w)}}{\Vert f\Vert _{X({\mathbb {R}},w)}}: f\in X_{\mathcal {S}}({\mathbb {R}},w) \right\} \Vert f\Vert _{X({\mathbb {R}},w)}, \end{aligned}$$

where

$$\begin{aligned} X_{\mathcal {S}}({\mathbb {R}},w):=({\mathcal {S}}({\mathbb {R}})\cap X({\mathbb {R}},w)){\setminus }\{0\}. \end{aligned}$$

Then, for every $\delta >0$,

$$\begin{aligned} \sup \left\{ \frac{\Vert {\mathcal {F}}^{-1}(a*\varphi _\delta ){\mathcal {F}}f\Vert _{X({\mathbb {R}},w)}}{\Vert f\Vert _{X({\mathbb {R}},w)}}: f\in X_{\mathcal {S}}({\mathbb {R}},w) \right\} \le \Vert a\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}. \end{aligned}$$

(2.6)

By Lemma 2.3(b), the Banach function space $X({\mathbb {R}},w)$ is separable. Then it follows from [1, Chap. 1, Corollary 5.6] and [24, Theorems 2.3 and 6.1] that for every $\delta >0$, the left-hand side of inequality (2.6) coincides with the multiplier norm $\Vert a*\varphi _\delta \Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}$, which completes the proof of inequality (2.5). $\square$

3 Continuous Fourier multipliers vanishing at infinity

3.1 The case of Lebesgue spaces with Muckenhoupt weights

The closure of a subset $\mathfrak{S}$ of a Banach space $\mathcal{E}$ in the norm of $\mathcal{E}$ will be denoted by ${\text {clos}}_{\mathcal {E}}({\mathfrak {S}})$.

Let $C_0({\mathbb {R}})$ be the set of all functions $f\in C({{\dot{{\mathbb {R}}}}})$ such that $f(\infty )=0$.

Lemma 3.1

Let $1<p<\infty$ and $w\in A_p({\mathbb {R}})$. Then

$$\begin{aligned} C_0({\mathbb {R}})\cap V({\mathbb {R}}) \subset {\text {clos}}_{{\mathcal {M}}_{L^p({\mathbb {R}},w)}}\big (C_c^\infty ({\mathbb {R}})\big ). \end{aligned}$$

Proof

The idea of the proof is borrowed from [20, Theorem 1.16] (see also [23, Theorem 3.1]). If $w\in A_p({\mathbb {R}})$, then $w^{1+\delta _2}\in A_{p(1+\delta _1)}({\mathbb {R}})$ whenever $|\delta _1|$ and $|\delta _2|$ are sufficiently small (see, e.g., [7, Theorem 2.31]). If $p\ge 2$, then one can find sufficiently small $\delta _1,\delta _2>0$ and a number $\theta \in (0,1)$ such that

$$\begin{aligned} \frac{1}{p}=\frac{1-\theta }{2}+\frac{\theta }{p(1+\delta _1)}, \quad w=1^{1-\theta }w^{(1+\delta _2)\theta }, \quad w^{1+\delta _2}\in A_{p(1+\delta _1)}({\mathbb {R}}). \end{aligned}$$

(3.1)

If $1<p<2$, then one can find a sufficiently small number $\delta _2>0$, a number $\delta _1<0$ with sufficiently small $|\delta _1|$, and a number $\theta \in (0,1)$ such that all conditions in (3.1) are fulfilled.

Let us use the following abbreviations:

$$\begin{aligned} {\mathcal {M}}_p:= & \, {\mathcal {M}}_{L^p({\mathbb {R}},w)}, \quad {\mathcal {M}}_{p_\theta }:={\mathcal {M}}_{L^{p(1+\delta _1)}({\mathbb {R}},w^{1+\delta _2})},\\ {\mathcal {B}}_p:= & \, {\mathcal {B}}(L^p({\mathbb {R}},w)), \quad {\mathcal {B}}_{p_\theta }:={\mathcal {B}}(L^{p(1+\delta _1)}({\mathbb {R}},w^{1+\delta _2})). \end{aligned}$$

For $n\in {\mathbb {N}}$, let

$$\begin{aligned} \psi _n(x):= \left\{ \begin{array}{lll} 1 &{} \quad \text{ if }& |x|\le n,\\ n+1-|x| & \quad \text{ if }&{} n<|x|<n+1,\\ 0 & \quad \text{ if }& |x|\ge n+1. \end{array}\right. \end{aligned}$$

(3.2)

Then $\psi _n$ has compact support and $\Vert \psi _n\Vert _{V({\mathbb {R}})}=3$. By the Stechkin-type inequality (1.4),

$$\begin{aligned} \Vert \psi _n\Vert _{{\mathcal {M}}_{p_\theta }}\le c_\theta , \end{aligned}$$

where $c_\theta$ is three times $c_{L^{p(1+\delta _1)}({\mathbb {R}},w^{1+\delta _2})}$, and the latter constant is the constant from (1.4).

Let $a\in \ C_0({\mathbb {R}})\cap V({\mathbb {R}})$. Fix $\varepsilon >0$. For $n\in {\mathbb {N}}$, take $b_n:=a\psi _n$. Then

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert a-b_n\Vert _{L^\infty ({\mathbb {R}})}=0 \end{aligned}$$

(3.3)

and $b_n\in C_0({\mathbb {R}})$ has compact support. Taking into account the Stechkin-type inequality (1.4), we get

$$\begin{aligned} \Vert a-b_n\Vert _{{\mathcal {M}}_{p_\theta }} \le \Vert a\Vert _{{\mathcal {M}}_{p_\theta }}(1+\Vert \psi _n\Vert _{{\mathcal {M}}_{p_\theta }}) \le (1+c_\theta )c_\theta \Vert a\Vert _{V({\mathbb {R}})} \end{aligned}$$

(3.4)

and

$$\begin{aligned} \Vert b_n\Vert _{{\mathcal {M}}_{p_\theta }} \le \Vert a\Vert _{{\mathcal {M}}_{p_\theta }}\Vert \psi _n\Vert _{{\mathcal {M}}_{p_\theta }} \le c_\theta ^2\Vert a\Vert _{V({\mathbb {R}})}. \end{aligned}$$

(3.5)

It follows from (3.1) and the Stein–Weiss interpolation theorem (see, e.g., [1, Chap. 3, Theorem 3.6]) that

$$\begin{aligned} \Vert a-b_n\Vert _{{\mathcal {M}}_p}&= \Vert W^0(a-b_n)\Vert _{{\mathcal {B}}_p} \nonumber \\&\le \Vert W^0(a-b_n)\Vert _{{\mathcal {B}}(L^2({\mathbb {R}}))}^{1-\theta } \Vert W^0(a-b_n)\Vert _{{\mathcal {B}}_{p_\theta }}^\theta \nonumber \\&= \Vert a-b_n\Vert _{L^\infty ({\mathbb {R}})}^{1-\theta } \Vert a-b_n\Vert _{{\mathcal {M}}_{p_\theta }}^\theta . \end{aligned}$$

(3.6)

Combining (3.3), (3.4) and (3.6), we see that there exists $n_0\in {\mathbb {N}}$ such that

$$\begin{aligned} \Vert a-b_{n_0}\Vert _{{\mathcal {M}}_p}<\varepsilon /2. \end{aligned}$$

(3.7)

Let $\varphi \in C_c^\infty ({\mathbb {R}})$ be a non-negative even function satisfying (2.3) and for $\delta >0$ let the function $\varphi _\delta$ be defined by (2.4). By Theorem 2.6 and inequality (3.5), for every $\delta >0$,

$$\begin{aligned} \Vert b_{n_0}*\varphi _\delta \Vert _{{\mathcal {M}}_{p_\theta }} \le \Vert b_{n_0}\Vert _{{\mathcal {M}}_{p_\theta }} \le c_\theta ^2\Vert a\Vert _{V({\mathbb {R}})}. \end{aligned}$$

(3.8)

It follows from [12, Propositions 4.18, 4.20–4.21] that $b_{n_0}*\varphi _\delta \in C_c^\infty ({\mathbb {R}})$ and

$$\begin{aligned} \lim _{\delta \rightarrow 0^+}\Vert b_{n_0}*\varphi _\delta -b_{n_0}\Vert _{L^\infty ({\mathbb {R}})}=0. \end{aligned}$$

(3.9)

In view of (3.1) and the Stein-Weiss interpolation theorem (see, e.g., [1, Chap. 3, Theorem 3.6]), we see that

$$\begin{aligned}&\Vert b_{n_0}*\varphi _\delta -b_{n_0}\Vert _{{\mathcal {M}}_p} \nonumber \\&\quad = \Vert W^0(b_{n_0}*\varphi _\delta -b_{n_0})\Vert _{{\mathcal {B}}_p} \nonumber \\&\quad \le \Vert W^0(b_{n_0}*\varphi _\delta -b_{n_0})\Vert _{{\mathcal {B}}(L^2({\mathbb {R}}))}^{1-\theta } \Vert W^0(b_{n_0}*\varphi _\delta -b_{n_0})\Vert _{{\mathcal {B}}_{p_\theta }}^\theta \nonumber \\&\quad = \Vert b_{n_0}*\varphi _\delta -b_{n_0}\Vert _{L^\infty ({\mathbb {R}})}^{1-\theta } \Vert b_{n_0}*\varphi _\delta -b_{n_0}\Vert _{{\mathcal {M}}_{p_\theta }}^\theta \nonumber \\&\quad \le \Vert b_{n_0}*\varphi _\delta -b_{n_0}\Vert _{L^\infty ({\mathbb {R}})}^{1-\theta } ( \Vert b_{n_0}*\varphi _\delta \Vert _{{\mathcal {M}}_{p_\theta }}+\Vert b_{n_0}\Vert _{{\mathcal {M}}_{p_\theta }} )^\theta . \end{aligned}$$

(3.10)

Combining (3.8)–(3.10), we conclude that there exists $\delta _0>0$ such that

$$\begin{aligned} \Vert b_{n_0}*\varphi _{\delta _0}-b_{n_0}\Vert _{{\mathcal {M}}_p}<\varepsilon /2. \end{aligned}$$

(3.11)

Hence, it follows from (3.7) and (3.11) that for every function a in the intersection $C_0({\mathbb {R}})\cap V({\mathbb {R}})$ and every $\varepsilon >0$ there exists a function $b_{n_0}*\varphi _{\delta _0}\in C_c^\infty ({\mathbb {R}})$ such that $\Vert a-b_{n_0}*\varphi _{\delta _0}\Vert _{{\mathcal {M}}_p}<\varepsilon$. Therefore, $a\in {\text {clos}}_{{\mathcal {M}}_p}\big (C_c^\infty ({\mathbb {R}})\big )$. $\square$

3.2 The case of rearrangement-invariant spaces with Muckenhoupt weights

The following lemma is an extension of the previous result to the case of rearrangement-invariant Banach function spaces with Muckenhoupt weights.

Lemma 3.2

Let $X({\mathbb {R}})$ be a separable rearrangement-invariant Banach function space with the Boyd indices satisfying $0<\alpha _X$, $\beta _X<1$. Suppose that a weight w belongs to $A_{1/\alpha _X}({\mathbb {R}})\cap A_{1/\beta _X}({\mathbb {R}})$. Then

$$\begin{aligned} C_0({\mathbb {R}})\cap V({\mathbb {R}}) \subset {\text {clos}}_{{\mathcal {M}}_{X({\mathbb {R}},w)}}\big (C_c^\infty ({\mathbb {R}})\big ). \end{aligned}$$

Proof

Since $\alpha _X,\beta _X\in (0,1)$ and $w\in A_{1/\alpha _X}({\mathbb {R}})\cap A_{1/\beta _X}({\mathbb {R}})$, it follows from [7, Theorem 2.31] that there exist p and q such that

$$\begin{aligned} 1<q<1/\beta _X\le 1/\alpha _X<p<\infty , \quad w\in A_p({\mathbb {R}})\cap A_q({\mathbb {R}}). \end{aligned}$$

(3.12)

Let $C_{p,q}\in (0,\infty )$ be the constant from estimate (2.1). Fix $\varepsilon >0$ and take a function $a\in C_0({\mathbb {R}})\cap V({\mathbb {R}})$. As in the proof of inequality (3.7) (see the proof of Lemma 3.1), it can be shown that there exists $n_0\in {\mathbb {N}}$ such that

$$\begin{aligned} \Vert a-b_{n_0}\Vert _{{\mathcal {M}}_{L^p({\mathbb {R}},w)}}<\frac{\varepsilon }{2C_{p,q}}, \quad \Vert a-b_{n_0}\Vert _{{\mathcal {M}}_{L^q({\mathbb {R}},w)}}<\frac{\varepsilon }{2C_{p,q}}, \end{aligned}$$

(3.13)

where $b_n=a\psi _n$ and $\psi _n$ is given by (3.2) for every $n\in {\mathbb {N}}$. It follows from (3.12), (3.13) and Theorem 2.1 that

$$\begin{aligned}&\Vert a-b_{n_0}\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}} = \Vert W^0(a-b_{n_0})\Vert _{{\mathcal {B}}(X({\mathbb {R}},w))} \nonumber \\&\quad = \Vert wW^0(a-b_{n_0})w^{-1}I\Vert _{{\mathcal {B}}(X({\mathbb {R}}))} \nonumber \\&\quad \le C_{p,q}\max \left\{ \Vert wW^0(a-b_{n_0})w^{-1}I\Vert _{{\mathcal {B}}(L^p({\mathbb {R}}))}, \Vert wW^0(a-b_{n_0})w^{-1}I\Vert _{{\mathcal {B}}(L^q({\mathbb {R}}))} \right\} \nonumber \\&\quad = C_{p,q}\max \left\{ \Vert W^0(a-b_{n_0})\Vert _{{\mathcal {B}}(L^p({\mathbb {R}},w))}, \Vert W^0(a-b_{n_0})\Vert _{{\mathcal {B}}(L^q({\mathbb {R}},w))} \right\} \nonumber \\&\quad = C_{p,q} \max \left\{ \Vert a-b_{n_0}\Vert _{{\mathcal {M}}_{L^p({\mathbb {R}},w)}}, \Vert a-b_{n_0}\Vert _{{\mathcal {M}}_{L^q({\mathbb {R}},w)}} \right\} <\varepsilon /2. \end{aligned}$$

(3.14)

As in the proof of inequality (3.11) (see the proof of Lemma 3.1), it can be shown that there exists $\delta _0>0$ such that

$$\begin{aligned} \Vert b_{n_0}*\varphi _{\delta _0}-b_{n_0}\Vert _{{\mathcal {M}}_{L^p({\mathbb {R}},w)}}<\frac{\varepsilon }{2C_{p,q}}, \quad \Vert b_{n_0}*\varphi _{\delta _0}-b_{n_0}\Vert _{{\mathcal {M}}_{L^q({\mathbb {R}},w)}}<\frac{\varepsilon }{2C_{p,q}}, \end{aligned}$$

(3.15)

where $\varphi \in C_c^\infty ({\mathbb {R}})$ is a non-negative even function satisfying (2.3) and the functions $\varphi _\delta$ are defined for all $\delta >0$ by (2.4). Arguing as in the proof of (3.14), we deduce from (3.12), (3.15) and Theorem 2.1 that

$$\begin{aligned} \Vert b_{n_0}*\varphi _{\delta _0}-b_{n_0}\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}<\varepsilon /2. \end{aligned}$$

(3.16)

It follows from (3.14) and (3.16) that for every function a in the intersection $C_0({\mathbb {R}})\cap V({\mathbb {R}})$ and every $\varepsilon >0$ there exists a function $b_{n_0}*\varphi _{\delta _0}\in C_c^\infty ({\mathbb {R}})$ such that $\Vert a-b_{n_0}*\varphi _{\delta _0}\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}<\varepsilon$. Therefore, $a\in {\text {clos}}_{{\mathcal {M}}_{X(R,w)}}\big (C_c^\infty ({\mathbb {R}})\big )$. $\square$

Now we are in a position to prove the main result of this section.

Theorem 3.3

Let $X({\mathbb {R}})$ be a separable rearrangement-invariant Banach function space with the Boyd indices satisfying $0<\alpha _X$, $\beta _X<1$. Suppose that a weight w belongs to $A_{1/\alpha _X}({\mathbb {R}})\cap A_{1/\beta _X}({\mathbb {R}})$. Consider the set

$$\begin{aligned} C_{0,X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}}):=\left\{ a\in C_{X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}})\ :\ a(\infty )=0\right\} . \end{aligned}$$

(3.17)

Then

$$\begin{aligned} C_{0,X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}}) = {\text {clos}}_{{\mathcal {M}}_{X({\mathbb {R}},w)}}\big (C_c^\infty ({\mathbb {R}})\big ). \end{aligned}$$

(3.18)

Proof

Let $a\in C_{X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}})$ be such that $a(\infty )=0$. Fix $\varepsilon >0$. By the definition of the algebra $C_{X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}})$, there exists a function $b\in C({{\dot{{\mathbb {R}}}}})\cap V({\mathbb {R}})$ such that

$$\begin{aligned} \Vert a-b\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}<\varepsilon /3. \end{aligned}$$

(3.19)

It follows from this observation and the continuous embedding of ${\mathcal {M}}_{X({\mathbb {R}},w)}$ into $L^\infty ({\mathbb {R}})$ (see Theorem 2.4) that

$$\begin{aligned} |b(\infty )|=|a(\infty )-b(\infty )| \le \Vert a-b\Vert _{L^\infty ({\mathbb {R}})}\le \Vert a-b\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}<\varepsilon /3. \end{aligned}$$

(3.20)

Take $c=b-b(\infty )\in C_0({\mathbb {R}})\cap V({\mathbb {R}})$. By Lemma 3.2, there exists a function $d\in C_c^\infty ({\mathbb {R}})\subset {\mathcal {M}}_{X({\mathbb {R}},w)}$ such that

$$\begin{aligned} \Vert c-d\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}<\varepsilon /3. \end{aligned}$$

(3.21)

Combining inequalities (3.19)–(3.21), we see that

$$\begin{aligned} \Vert a-d\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}} \le \Vert a-b\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}} + |b(\infty )| + \Vert c-d\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}<\varepsilon . \end{aligned}$$

Hence

$$\begin{aligned} C_{0,X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}}) \subset {\text {clos}}_{{\mathcal {M}}_{X({\mathbb {R}},w)}}\big (C_c^\infty ({\mathbb {R}})\big ). \end{aligned}$$

(3.22)

Let us prove the reverse embedding. Take $a\in {\text {clos}}_{{\mathcal {M}}_{X({\mathbb {R}},w)}}\big (C_c^\infty ({\mathbb {R}})\big )$. Then there exists a sequence $\{a_n\}_{n\in {\mathbb {N}}}\subset C_c^\infty ({\mathbb {R}})$ such that

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert a_n-a\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}=0. \end{aligned}$$

Since $C_c^\infty ({\mathbb {R}})\subset C({{\dot{{\mathbb {R}}}}})\cap V({\mathbb {R}})$, the above equality and the continuous embedding of the algebra ${\mathcal {M}}_{X({\mathbb {R}},w)}$ into the algebra $L^\infty ({\mathbb {R}})$ (see Theorem 2.4) imply that $a\in C_{X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}})$ and

$$\begin{aligned} |a(\infty )|&= \lim _{n\rightarrow \infty }|a_n(\infty )-a(\infty )| \le \lim _{n\rightarrow \infty }\Vert a_n-a\Vert _{L^\infty ({\mathbb {R}})} \\&\le \lim _{n\rightarrow \infty }\Vert a_n-a\Vert _{{\mathcal {M}}_{X(R,w)}}=0. \end{aligned}$$

Thus

$$\begin{aligned} {\text {clos}}_{{\mathcal {M}}_{X({\mathbb {R}},w)}}\big (C_c^\infty ({\mathbb {R}})\big ) \subset C_{0,X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}}). \end{aligned}$$

(3.23)

Combining (3.22) and (3.23), we arrive at (3.18). $\square$

4 Continuous and slowly oscillating Fourier multipliers

4.1 Continuous Fourier multipliers on one and two-point compactifications of the real line

For a function $f\in C({\overline{{\mathbb {R}}}})$, let

$$\begin{aligned} J_f(x):=\left\{ \begin{array}{lll} f(-\infty ) &{} \quad \text{ if }&{} x\in (-\infty ,-1), \\ \frac{1}{2}\big [f(-\infty )(1-x)+f(+\infty )(1+x)\big ] &{} \quad \text{ if } &{} x\in [-1,1], \\ f(+\infty ) &{} \quad \text{ if }&{} x\in (1,+\infty ). \end{array}\right. \end{aligned}$$

(4.1)

It is easy to see that

$$\begin{aligned} \Vert J_f\Vert _{V({\mathbb {R}})}=\max \big \{|f(-\infty )|,|f(+\infty )|\big \}+ |f(+\infty )-f(-\infty )|. \end{aligned}$$

(4.2)

Therefore $J_f\in C({\overline{{\mathbb {R}}}})\cap V({\mathbb {R}})$ and $f-J_f\in C_0({\mathbb {R}})$.

The next lemma extends [29, Lemma 3.1(i)] from the setting of Lebesgue spaces to the setting of rearrangement-invariant Banach function spaces with Muckenhoupt weights.

Lemma 4.1

Let $X({\mathbb {R}})$ be a separable rearrangement-invariant Banach function space with the Boyd indices satisfying $0<\alpha _X$, $\beta _X<1$. Suppose that $w\in A_{1/\alpha _X}({\mathbb {R}})\cap A_{1/\beta _X}({\mathbb {R}})$. Then

$$\begin{aligned} C_{X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}})=C_{X({\mathbb {R}},w)}({\overline{{\mathbb {R}}}})\cap C({{\dot{{\mathbb {R}}}}}). \end{aligned}$$

(4.3)

Proof

The proof is analogous to the proof of [29, Lemma 3.1(i)] (see also [23, Lemma 3.2]). It is obvious that $C_{X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}})\subset C_{X({\mathbb {R}},w)}({\overline{{\mathbb {R}}}})$. On the other hand, it follows from Theorem 2.4 that $C_{X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}})\subset C({{\dot{{\mathbb {R}}}}})$. Therefore,

$$\begin{aligned} C_{X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}})\subset C_{X({\mathbb {R}},w)}({\overline{{\mathbb {R}}}})\cap C({{\dot{{\mathbb {R}}}}}). \end{aligned}$$

(4.4)

To prove the opposite embedding, let us consider an arbitrary function $a\in C_{X({\mathbb {R}},w)}({\overline{{\mathbb {R}}}})$ such that $a(+\infty )=a(-\infty )$. Let $\{a_n\}_{n\in {\mathbb {N}}}\subset C({\overline{{\mathbb {R}}}})\cap V({\mathbb {R}})$ be a sequence such that $\Vert a_n-a\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}\rightarrow 0$ as $n\rightarrow \infty$. According to Theorem 2.4, the sequence $\{a_n\}_{n\in {\mathbb {N}}}$ converges to a uniformly on ${\mathbb {R}}$. Hence, in particular, $a_n(\pm \infty )\rightarrow a(\infty )$ as $n\rightarrow \infty$. Let the functions $b_n:=J_{a_n-a(\infty )}$ be defined by (4.1) with $a_n-a(\infty )$ in place of f. By the Stechkin-type inequality (1.4) and equality (4.2), we have

$$\begin{aligned} \Vert b_n\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}&\le c_{X({\mathbb {R}},w)}\Vert J_{a_n-a(\infty )}\Vert _{V({\mathbb {R}})} \\&= c_{X({\mathbb {R}},w)}\max \big \{|a_n(-\infty )-a(\infty )|,|a_n(+\infty )-a(\infty )|\big \} \\&\quad +c_{X({\mathbb {R}},w)}|a_n(+\infty )-a_n(-\infty )|. \end{aligned}$$

Therefore, $\Vert b_n\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}\rightarrow 0$ as $n\rightarrow \infty$ and thus,

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert a_n-b_n-a\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}=0. \end{aligned}$$

Since $a_n-b_n\in C({{\dot{{\mathbb {R}}}}})\cap V({\mathbb {R}})$, the latter equality implies that $a\in C_{X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}})$. Thus

$$\begin{aligned} C_{X({\mathbb {R}},w)}({\overline{{\mathbb {R}}}})\cap C({{\dot{{\mathbb {R}}}}})\subset C_{X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}}). \end{aligned}$$

(4.5)

Combining embeddings (4.4)–(4.5), we arrive at equality (4.3). $\square$

4.2 Embedding of the algebra $C_{X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}})$ into the algebra $SO_{X({\mathbb {R}},w)}$ of slowly oscillating Fourier multipliers

Let $C_b({\mathbb {R}}):=C({\mathbb {R}})\cap L^\infty ({\mathbb {R}})$. For a bounded measurable function $f:{\mathbb {R}}\rightarrow {\mathbb {C}}$ and a set $J\subset {\mathbb {R}}$, let

$$\begin{aligned} {\text {osc}}(f,J):= \mathop {\text {ess sup}}\limits _{x,y\in J}|f(x)-f(y)|. \end{aligned}$$

Let SO be the $C^*$-algebra of all slowly oscillating functions at $\infty$ defined by

$$\begin{aligned} SO:=\left\{ f\in C_b({\mathbb {R}}): \lim _{x\rightarrow +\infty } {\text {osc}}(f,[-x,-x/2]\cup [x/2,x])=0\right\} . \end{aligned}$$

Consider the differential operator $(Df)(x)=xf'(x)$ and its iterations defined by $D^0f=f$ and $D^jf=D(D^{j-1}f)$ for $j\in {\mathbb {N}}$. Let

$$\begin{aligned} SO^3:=\left\{ a\in SO\cap C^3({\mathbb {R}}):\lim _{x\rightarrow \infty } (D^ja)(x)=0,\ j=1,2,3 \right\} , \end{aligned}$$

where $C^3({\mathbb {R}})$ denotes the set of all three times continuously differentiable functions. It is easy to see that $SO^3$ is a commutative Banach algebra under pointwise operations and the norm

$$\begin{aligned} \Vert a\Vert _{SO^3}:=\sum _{j=0}^3\frac{1}{j!}\Vert D^ja\Vert _{L^\infty ({\mathbb {R}})}. \end{aligned}$$

It follows from [21, Corollary 2.6] that if $X({\mathbb {R}})$ is a separable rearrangement-invariant Banach function space with the Boyd indices $\alpha _X,\beta _X$ such that $0<\alpha _X$, $\beta _X<1$ and $w\in A_{1/\alpha _X}({\mathbb {R}})\cap A_{1/\beta _X}({\mathbb {R}})$, then there exists a constant $c_{X({\mathbb {R}},w)}\in (0,\infty )$ such that for all $a\in SO^3$,

$$\begin{aligned} \Vert a\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}\le c_{X({\mathbb {R}},w)}\Vert a\Vert _{SO^3}. \end{aligned}$$

The continuous embedding $SO^3\subset {\mathcal {M}}_{X({\mathbb {R}},w)}$ allows us to define the algebra $SO_{X({\mathbb {R}},w)}$ of slowly oscillating Fourier multipliers as the closure of $SO^3$ with respect to the multiplier norm:

$$\begin{aligned} SO_{X({\mathbb {R}},w)}:={\text {clos}}_{{\mathcal {M}}_{X({\mathbb {R}},w)}}\big (SO^3\big ). \end{aligned}$$

The following result is analogous to [28, Lemma 3.6].

Lemma 4.2

Let $X({\mathbb {R}})$ be a separable rearrangement-invariant Banach function space with the Boyd indices satisfying $0<\alpha _X$, $\beta _X<1$. Suppose that a weight w belongs to $A_{1/\alpha _X}({\mathbb {R}})\cap A_{1/\beta _X}({\mathbb {R}})$. Then $C_{X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}})\subset SO_{X({\mathbb {R}},w)}$.

Proof

Let $a\in C_{X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}})$. Fix $\varepsilon >0$. Then there exists $b\in C({{\dot{{\mathbb {R}}}}})\cap V({\mathbb {R}})$ such that

$$\begin{aligned} \Vert a-b\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}<\varepsilon /2. \end{aligned}$$

(4.6)

Then $b-b(\infty )\in C_0({\mathbb {R}})\cap V({\mathbb {R}})$. By Lemma 3.2,

$$\begin{aligned} b-b(\infty )\in {\text {clos}}_{{\mathcal {M}}_{X({\mathbb {R}},w)}} \big (C_c^\infty ({\mathbb {R}})\big ). \end{aligned}$$

Then there exists $c\in C_c^\infty ({\mathbb {R}})$ such that

$$\begin{aligned} \Vert b-b(\infty )-c\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}<\varepsilon /2. \end{aligned}$$

(4.7)

It follows from inequalities (4.6) and (4.7) that

$$\begin{aligned} \Vert a-(c+b(\infty ))\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}<\varepsilon . \end{aligned}$$

Since $c+b(\infty )\in C_c^\infty ({\mathbb {R}})\dot{+}{\mathbb {C}}\subset SO^3$, we get $a\in {\text {clos}}_{{\mathcal {M}}_{X({\mathbb {R}},w)}}(SO^3)=SO_{X({\mathbb {R}},w)}$. $\square$

5 Almost periodic Fourier multipliers and their products with continuous Fourier multipliers vanishing at infinity

5.1 The algebra $AP_{X({\mathbb {R}},w)}$ of almost periodic Fourier multipliers

For $\lambda \in {\mathbb {R}}$, let $T_\lambda$ denote the translation operator defined by

$$\begin{aligned} (T_\lambda f)(x)=f(x-\lambda ),\quad x\in {\mathbb {R}}. \end{aligned}$$

Lemma 5.1

Let $X({\mathbb {R}})$ be a rearrangement-invariant Banach function space and $w:{\mathbb {R}}\rightarrow [0,\infty ]$ be a weight such that $w\in X_{\mathrm{loc}}({\mathbb {R}})$ and $1/w\in X_{\mathrm{loc}}'({\mathbb {R}})$. Suppose that $\lambda \in {\mathbb {R}}$. Then the translation operator $T_\lambda$ is bounded on the Banach function space $X({\mathbb {R}},w)$ if and only if the function

$$\begin{aligned} v_\lambda (x):=\frac{w(x+\lambda )}{w(x)},\quad x\in {\mathbb {R}}, \end{aligned}$$

belongs to the space $L^\infty ({\mathbb {R}})$. In that case $\Vert T_\lambda \Vert _{{\mathcal {B}}(X({\mathbb {R}},w))}=\Vert v_\lambda \Vert _{L^\infty ({\mathbb {R}})}$.

Proof

The operator $T_\lambda$ is bounded on the space $X({\mathbb {R}},w)$ if and only if the operator $wT_\lambda w^{-1}I=T_\lambda (v_\lambda I)$ is bounded on the space $X({\mathbb {R}})$. Moreover, their norms coincide. It is easy to see that for every $f\in X({\mathbb {R}})$, the function $T_\lambda f$ is equimeasurable with f, whence $\Vert T_\lambda f\Vert _{X({\mathbb {R}})}=\Vert f\Vert _{X({\mathbb {R}})}$. Therefore,

$$\begin{aligned} \Vert T_\lambda \Vert _{{\mathcal {B}}(X({\mathbb {R}},w))} = \Vert T_\lambda (v_\lambda I)\Vert _{{\mathcal {B}}(X({\mathbb {R}}))} = \Vert v_\lambda I\Vert _{{\mathcal {B}}(X({\mathbb {R}}))}. \end{aligned}$$

By [31, Theorem 1], the multiplication operator $v_\lambda I$ is bounded on the space $X({\mathbb {R}})$ if and only if $v_\lambda \in L^\infty ({\mathbb {R}})$ and $\Vert v_\lambda I\Vert _{{\mathcal {B}}(X({\mathbb {R}}))}=\Vert v_\lambda \Vert _{L^\infty ({\mathbb {R}})}$. Thus, $\Vert T_\lambda \Vert _{{\mathcal {B}}(X({\mathbb {R}},w))}=\Vert v_\lambda \Vert _{L^\infty ({\mathbb {R}})}$. $\square$

As a consequence of the previous result, we show that for all $\lambda \in {\mathbb {R}}$, the exponential functions $e_\lambda (x)=e^{i\lambda x}$, $x\in {\mathbb {R}}$, are Fourier multipliers on separable rearrangement-invariant Banach function spaces with weights in the sublclass $A_{1/\alpha _X}^0({\mathbb {R}})\cap A_{1/\beta _X}^0({\mathbb {R}})$ of the class of Muckenhoupt weights.

Corollary 5.2

Let $X({\mathbb {R}})$ be a separable rearrangement-invariant Banach function space with the Boyd indices satisfying $0<\alpha _X\le \beta _X<1$. Suppose that a weight w belongs to $A_{1/\alpha _X}^0({\mathbb {R}})\cap A_{1/\beta _X}^0({\mathbb {R}})$. Then for every $\lambda \in {\mathbb {R}}$, the function $e_\lambda$ belongs to ${\mathcal {M}}_{X({\mathbb {R}},w)}$ and $\Vert e_\lambda \Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}=\Vert v_\lambda \Vert _{L^\infty ({\mathbb {R}})}$.

Proof

It follows from the definition of the classes $A_{1/\alpha _X}^0({\mathbb {R}})$ and $A_{1/\beta _X}^0({\mathbb {R}})$ that the function $v_\lambda (x)=\frac{w(x+\lambda )}{w(x)}$, $x\in {\mathbb {R}}$, is bounded for every $\lambda \in {\mathbb {R}}$. By Lemma 2.3(a), $w\in X_{\mathrm{loc}}({\mathbb {R}})$ and $1/w\in X_{\mathrm{loc}}'({\mathbb {R}})$. Then, by Lemma 5.1, the operator $T_\lambda$ is bounded on the Banach function space $X({\mathbb {R}},w)$ and

$$\begin{aligned} \Vert T_\lambda \Vert _{{\mathcal {B}}(X({\mathbb {R}},w))}=\Vert v_\lambda \Vert _{L^\infty ({\mathbb {R}})}, \quad \lambda \in {\mathbb {R}}. \end{aligned}$$

It remains to observe that $T_\lambda =W^0(e_\lambda )$. Thus $e_\lambda \in {\mathcal {M}}_{X({\mathbb {R}},w)}$ and

$$\begin{aligned} \Vert e_\lambda \Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}} = \Vert W^0(e_\lambda )\Vert _{{\mathcal {B}}(X({\mathbb {R}},w))} = \Vert v_\lambda \Vert _{L^\infty ({\mathbb {R}})}, \quad \lambda \in {\mathbb {R}}, \end{aligned}$$

which completes the proof. $\square$

Corollary 5.2 implies that if $X({\mathbb {R}})$ is a separable rearrangement-invariant Banach function spaces and $w\in A_{1/\alpha _X}^0({\mathbb {R}})\cap A_{1/\beta _X}^0({\mathbb {R}})$, then $APP\subset {\mathcal {M}}_{X({\mathbb {R}},w)}$. We define the algebra $AP_{X({\mathbb {R}},w)}$ of almost periodic Fourier multipliers by

$$\begin{aligned} AP_{X({\mathbb {R}},w)}:={\text {clos}}_{{\mathcal {M}}_{X({\mathbb {R}},w)}}\big (APP\big ). \end{aligned}$$

It is natural to refer to the weights in $A_{1/\alpha _X}^0\cap A_{1/\beta _X}^0$ as suitable Muckenhoupt weights. The class of suitable Muckenhoput weights contains many nontrivial weights as the following example shows.

For $\delta ,\nu ,\eta \in {\mathbb {R}}$, consider the weight

$$\begin{aligned} w(x):=\left\{ \begin{array}{lll} \exp \big ( \delta +\nu \sin (\eta \log (\log |x|)) \big ) &{} \quad \text{ if }&{} |x|\ge e, \\ \exp (\delta ) &{} \quad \text{ if }&{} |x|<e. \end{array}\right. \end{aligned}$$

Let $r\in (1,\infty )$. It was shown in [27, Example 4.2] that if

$$\begin{aligned} -1/r<\delta -|\nu |\sqrt{\eta ^2+1} \le \delta +|\nu |\sqrt{\eta ^2+1}<1-1/r, \end{aligned}$$

then $w\in A_r^0({\mathbb {R}})$. Hence if $0<\alpha _X\le \beta _X<1$ and

$$\begin{aligned} -\alpha _X<\delta -|\nu |\sqrt{\eta ^2+1} \le \delta +|\nu |\sqrt{\eta ^2+1}<1-\beta _X, \end{aligned}$$

then $w\in A_{1/\alpha _X}^0({\mathbb {R}})\cap A_{1/\beta _X}^0({\mathbb {R}})$.

5.2 Products of almost periodic Fourier multipliers and continuous Fourier multipliers vanishing at infinity

The next lemma generalizes [29, Lemma 3.1(iii)] from the setting of Lebesgue spaces to the setting of rearrangement-invariant Banach function spaces with suitable Muckenhoupt weights.

Lemma 5.3

Let $X({\mathbb {R}})$ be a separable rearrangement-invariant Banach function space with the Boyd indices satisfying $0<\alpha _X$, $\beta _X<1$. Suppose that w belongs to $A_{1/\alpha _X}^0({\mathbb {R}})\cap A_{1/\beta _X}^0({\mathbb {R}})$ and $C_{0,X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}})$ is defined by (3.17). If $a\in AP_{X({\mathbb {R}},w)}$ and $\psi \in C_{0,X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}})$, then $a\psi \in C_{0,X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}})$.

Proof

By Theorem 3.3, there exists a sequence $\{\psi _n\}_{n\in {\mathbb {N}}}\subset C_c^\infty ({\mathbb {R}})$ such that

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert \psi _n-\psi \Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}=0. \end{aligned}$$

(5.1)

By the definition of the algebra $AP_{X({\mathbb {R}},w)}$, there exists a sequence $a_n\in APP$ such that

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert a_n-a\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}=0. \end{aligned}$$

(5.2)

Then $a_n\psi _n\in C_c^\infty ({\mathbb {R}})\subset C({{\dot{{\mathbb {R}}}}})\cap V({\mathbb {R}})$ for every $n\in {\mathbb {N}}$. Moreover, (5.1)–(5.2) imply that

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert a_n\psi _n-a\psi \Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}=0. \end{aligned}$$

Hence $a\psi \in C_{X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}})$. In view of the continuous embedding of ${\mathcal {M}}_{X({\mathbb {R}},w)}$ into $L^\infty ({\mathbb {R}})$ (see Theorem 2.4) and the above equality, we obtain

$$\begin{aligned} |(a\psi )(\infty )|&= \lim _{n\rightarrow \infty }|(a_n\psi _n)(\infty )-(a\psi )(\infty )| \le \lim _{n\rightarrow \infty }\Vert a_n\psi _n-a\psi \Vert _{L^\infty ({\mathbb {R}})} \\&\le \lim _{n\rightarrow \infty }\Vert a_n\psi _n-a\psi \Vert _{{\mathcal {M}}_{X(R,w)}}=0. \end{aligned}$$

Thus $(a\psi )(\infty )=0$ and $a\psi \in C_{0,X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}})$. $\square$

6 Proof of the main result

6.1 The algebra ${\mathcal {A}}_u$

For a real-valued monotonically increasing function $u\in C({\overline{{\mathbb {R}}}})$ such that

$$\begin{aligned} u(-\infty )=0 \quad u(+\infty )=1, \end{aligned}$$

(6.1)

consider the set

$$\begin{aligned} {\mathcal {A}}_u:= \left\{ a=(1-u)a_l+ua_r+a_0\ :\ a_l,a_r\in AP_{X({\mathbb {R}},w)},\ a_0\in C_{0,X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}}) \right\} . \end{aligned}$$

Lemma 6.1

Let $X({\mathbb {R}})$ be a separable rearrangement-invariant Banach function space with the Boyd indices satisfying $0<\alpha _X$, $\beta _X<1$. Suppose that $w\in A_{1/\alpha _X}^0({\mathbb {R}})\cap A_{1/\beta _X}^0({\mathbb {R}})$. If $u\in C({\overline{{\mathbb {R}}}})$ is a real-valued monotonically increasing function such that $u(-\infty )=0$ and $u(+\infty )=1$, then the set ${\mathcal {A}}_u$ is an algebra and the mappings $a\mapsto a_l$ and $a\mapsto a_r$ are algebraic homomorphisms of ${\mathcal {A}}_u$ onto $AP_{X({\mathbb {R}},w)}$.

Proof

If $a,b\in {\mathcal {A}}_u$, then

$$\begin{aligned} a=(1-u)a_l+ua_r+a_0, \quad b=(1-u)b_l+ub_r+b_0 \end{aligned}$$

with some $a_l,a_r,b_l,b_r\in AP_{X({\mathbb {R}},w)}$ and $a_0,b_0\in C_{0,X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}})$. Therefore

$$\begin{aligned} a+b=(1-u)(a_l+b_l)+u(a_r+b_r)+(a_0+b_0) \end{aligned}$$

(6.2)

and

$$\begin{aligned} ab&=(1-u)^2a_lb_l+u^2a_rb_r+(1-u)u(a_lb_r+a_rb_l) \nonumber \\&\quad + \big ((1-u)a_l+ua_r\big )b_0+ \big ((1-u)b_l+ub_r\big )a_0+a_0b_0 \nonumber \\&= (1-u)a_lb_l+ua_rb_r+c_0, \end{aligned}$$

(6.3)

where

$$\begin{aligned} c_0&= (u-u^2)\big [(a_lb_r+a_rb_l)-(a_lb_l+a_rb_r)\big ] \nonumber \\&\quad + \big ((1-u)a_l+ua_r\big )b_0+ \big ((1-u)b_l+ub_r\big )a_0+a_0b_0. \end{aligned}$$

(6.4)

Since $1-u,u\in C({\overline{{\mathbb {R}}}})\cap V({\mathbb {R}})\subset C_{X({\mathbb {R}},w)}({\overline{{\mathbb {R}}}})$ and $a_0,b_0\in C_{0,X({\mathbb {R}},w)}({\mathbb {R}})$, it follows from Lemma 4.1 that

$$\begin{aligned} (1-u)a_0, ua_0,(1-u)b_0,ub_0\in C_{0,X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}}). \end{aligned}$$

Then, by Lemma 5.3,

$$\begin{aligned} (1-u)a_lb_0,ua_rb_0,(1-u)b_la_0,ub_ra_0\in C_{0,X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}}). \end{aligned}$$

(6.5)

Since $u-u^2\in C({\overline{{\mathbb {R}}}})\cap V({\mathbb {R}})\subset C_{X({\mathbb {R}},w)}({\overline{{\mathbb {R}}}})$ and $u(\pm \infty )-u^2(\pm \infty )=0$, by Lemma 4.1, $u-u^2\in C_{0,X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}})$. Then, in view of Lemma 5.3, we also conclude that

$$\begin{aligned} (u-u^2)\big [(a_lb_r+a_rb_l)-(a_lb_l+a_rb_r)\big ] \in C_{0,X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}}). \end{aligned}$$

(6.6)

It follows from (6.4) to (6.6) that $c_0\in C_{0,X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}})$. In view of this observation and equalities (6.2)–(6.3), we see that $a+b,ab\in {\mathcal {A}}_u$. Therefore, ${\mathcal {A}}_u$ is an algebra. It is clear that the mappings $a\mapsto a_l$ and $a\mapsto a_r$ are algebraic homomorphisms of ${\mathcal {A}}_u$ onto $AP_{X({\mathbb {R}},w)}$. $\square$

6.2 The multiplier norm of $a=(1-u)a_r +ua_r+a_0\in {\mathcal {A}}_u$ dominates the multiplier norms of $a_r$ and $a_l$

In this section we will prepare the proof of the fact that the algebraic homomorphisms ${\mathcal {A}}_u\rightarrow AP_{X({\mathbb {R}},w)}$ given by $a\mapsto a_l$ and $a\mapsto a_r$ are actually Banach algebra homomorphisms of norm 1. To this end, we will show that for $a\in {\mathcal {A}}_u$,

$$\begin{aligned} \Vert a_r\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}\le \Vert a\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}, \quad \Vert a_l\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}\le \Vert a\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}. \end{aligned}$$

(6.7)

For $a\in L^\infty ({\mathbb {R}})$ and $h\in {\mathbb {R}}$, we define

$$\begin{aligned} a^h(x):=a(x+h), \quad x\in {\mathbb {R}}. \end{aligned}$$

The following consequence of Kronecker’s theorem (see, e.g., [10, Theorem 1.12]) plays a crucial role in the proof of inequalities (6.7).

Lemma 6.2

If $a_1,\ldots ,a_k\in APP$ is a finite collection of almost periodic polynomials, then there exists a sequence $\{h_n\}_{n\in {\mathbb {N}}}$ of real numbers such that $h_n\rightarrow +\infty$ as $n\rightarrow \infty$ and

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert a_m^{\pm h_n}-a_m\Vert _{L^\infty ({\mathbb {R}})}=0 \end{aligned}$$

for each $m\in \{1,\ldots ,k\}$.

For the sign “+”, the proof of the above lemma is given in [10, Lemma 10.2], for the sign “–”, the proof is analogous.

We start the proof of inequalities (6.7) for $a=(1-v)a_l+va_r+a_0$ with a nice function v in place of u and nice functions $a_l,a_r$ and $a_0$.

Lemma 6.3

Let $X({\mathbb {R}})$ be a separable rearrangement-invariant Banach function space with the Boyd indices satisfying $0<\alpha _X$, $\beta _X<1$. Suppose that $w\in A_{1/\alpha _X}^0({\mathbb {R}})\cap A_{1/\beta _X}^0({\mathbb {R}})$. Let $v\in C({\overline{{\mathbb {R}}}})$ be any real-valued monotonically increasing function such that there exists a point $x_0>0$ such that $v(x)=0$ for $x<-x_0$ and $v(x)=1$ for $x>x_0$. If $a_l,a_r\in APP$, $a_0\in C_c^\infty ({\mathbb {R}})$, and

$$\begin{aligned} a=(1-v)a_l+va_r+a_0, \end{aligned}$$

(6.8)

then inequalities (6.7) hold.

Proof

The idea of the proof is borrowed from the proof of [27, Theorem 3.1]. By Lemma 6.2, there is a sequence $\{h_n\}_{n\in {\mathbb {N}}}$ of real numbers such that $h_n\rightarrow +\infty$ as $n\rightarrow \infty$ and

$$\begin{aligned}&\lim _{n\rightarrow \infty }\Vert a_r^{h_n}-a_r\Vert _{L^\infty ({\mathbb {R}})}=0, \quad \lim _{n\rightarrow \infty }\Vert (a_r')^{h_n}-a_r'\Vert _{L^\infty ({\mathbb {R}})}=0, \end{aligned}$$

(6.9)

$$\begin{aligned}&\lim _{n\rightarrow \infty }\Vert a_l^{-h_n}-a_l\Vert _{L^\infty ({\mathbb {R}})}=0, \quad \lim _{n\rightarrow \infty }\Vert (a_l')^{-h_n}-a_l'\Vert _{L^\infty ({\mathbb {R}})}=0. \end{aligned}$$

(6.10)

Let us show that

$$\begin{aligned} \mathop {\text {s-lim}}\limits _{n\rightarrow \infty }e_{h_n}W^0(a)e_{-h_n}I=W^0(a_r), \quad \mathop {\text {s-lim}}\limits _{n\rightarrow \infty }e_{-h_n}W^0(a)e_{h_n}I=W^0(a_l) \end{aligned}$$

(6.11)

on the space $X({\mathbb {R}},w)$. As

$$\begin{aligned} e_{\pm h_n}W^0(a)e_{\mp h_n}I=W^0(a^{\pm h_n}), \end{aligned}$$

we have to prove that for every $f\in X({\mathbb {R}},w)$,

$$\begin{aligned}&\lim _{n\rightarrow \infty }\left\| W^0(a^{h_n}-a_r)f\right\| _{X({\mathbb {R}},w)}=0, \end{aligned}$$

(6.12)

$$\begin{aligned}&\lim _{n\rightarrow \infty }\left\| W^0(a^{-h_n}-a_l)f\right\| _{X({\mathbb {R}},w)}=0. \end{aligned}$$

(6.13)

Since the operators $W^0(a^{h_n}-a_r)$ and $W^0(a^{-h_n}-a_l)$ are uniformly bounded in $n\in {\mathbb {N}}$ and the set ${\mathcal {S}}_0({\mathbb {R}})$ is dense in the space $X({\mathbb {R}},w)$ in view of Lemma 2.3, applying [34, Lemma 1.4.1], we conclude that it is enough to prove equalities (6.12)–(6.13) for all $f\in {\mathcal {S}}_0({\mathbb {R}})$.

Fix $f\in {\mathcal {S}}_0({\mathbb {R}})$. Then, by a smooth version of Urysohn’s lemma (see, e.g., [17, Proposition 6.5]), there is a function $\psi \in C_c^\infty ({\mathbb {R}})$ such that $0\le \psi \le 1$, ${\text {supp}}{\mathcal {F}}f\subset {\text {supp}}\psi$ and $\psi |_{{\text {supp}}{\mathcal {F}}f}=1$. Therefore, for all $n\in {\mathbb {N}}$,

$$\begin{aligned} W^0(a^{h_n}-a_r)f={\mathcal {F}}^{-1}(a^{h_n}-a_r)\psi {\mathcal {F}}f, \quad W^0(a^{-h_n}-a_l)f={\mathcal {F}}^{-1}(a^{-h_n}-a_l)\psi {\mathcal {F}}f \end{aligned}$$

and

$$\begin{aligned}&\left\| W^0(a^{h_n}-a_r)f\right\| _{X({\mathbb {R}},w)} \le \Vert (a^{h_n}-a_r)\psi \Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}\Vert f\Vert _{X({\mathbb {R}},w)}, \end{aligned}$$

(6.14)

$$\begin{aligned}&\left\| W^0(a^{-h_n}-a_l)f\right\| _{X({\mathbb {R}},w)} \le \Vert (a^{-h_n}-a_l)\psi \Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}\Vert f\Vert _{X({\mathbb {R}},w)}. \end{aligned}$$

(6.15)

Since $v(x)=1$ for $x>x_0$ and $v(x)=0$ for $x<-x_0$ and $a_0\in C_c^\infty ({\mathbb {R}})$, there exists $N\in {\mathbb {N}}$ such that for all $x\in {\text {supp}}\psi$ and $n>N$,

$$\begin{aligned} v(x+h_n)=1, \quad v(x-h_n)=0, \quad a_0(x\pm h_n)=0. \end{aligned}$$

Hence, for all $n>N$ and $x\in {\mathbb {R}}$,

$$\begin{aligned} \big (a^{h_n}(x)-a_r(x)\big )\psi (x)&= \big (a_r^{h_n}(x)-a_r(x)\big )\psi (x), \end{aligned}$$

(6.16)

$$\begin{aligned} \big (a^{-h_n}(x)-a_l(x)\big )\psi (x)&= \big (a_l^{-h_n}(x)-a_l(x)\big )\psi (x). \end{aligned}$$

(6.17)

It is clear that the functions on the right-hand sides of (6.16)–(6.17) belong to $C_c^\infty ({\mathbb {R}})$. Therefore, by the Stechkin-type inequality (1.4), for all $n>N$,

$$\begin{aligned} \big \Vert \big (a^{h_n}-a_r\big )\psi \big \Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}&= \big \Vert \big (a_r^{h_n}-a_r\big )\psi \big \Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}} \nonumber \\&\le c_{X({\mathbb {R}},w)}\big \Vert \big (a_r^{h_n}-a_r\big )\psi \big \Vert _{V({\mathbb {R}})} \nonumber \\&= c_{X({\mathbb {R}},w)}\big \Vert \big (a_r^{h_n}-a_r\big )\psi \big \Vert _{L^\infty ({\mathbb {R}})} \nonumber \\&\quad + c_{X({\mathbb {R}},w)}\int _{\mathbb {R}}\left| (a_r^{h_n})'(x)-a_r'(x)\right| \,|\psi (x)|\,dx \nonumber \\&\quad + c_{X({\mathbb {R}},w)}\int _{\mathbb {R}}\left| a_r^{h_n}(x)-a_r(x)\right| \,|\psi '(x)|\,dx \nonumber \\&\le c_{X({\mathbb {R}},w)}\big (\Vert \psi \Vert _{L^\infty ({\mathbb {R}})}+\Vert \psi '\Vert _{L^1({\mathbb {R}})}\big ) \big \Vert a_r^{h_n}-a_r\big \Vert _{L^\infty ({\mathbb {R}})} \nonumber \\&\quad + c_{X({\mathbb {R}},w)}\Vert \psi \Vert _{L^1({\mathbb {R}})} \big \Vert (a_r^{h_n})'-a_r'\big \Vert _{L^\infty ({\mathbb {R}})} \end{aligned}$$

(6.18)

and, analogously,

$$\begin{aligned} \big \Vert \big (a^{-h_n}-a_l\big )\psi \big \Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}&\le c_{X({\mathbb {R}},w)}\big (\Vert \psi \Vert _{L^\infty ({\mathbb {R}})}+\Vert \psi '\Vert _{L^1({\mathbb {R}})}\big ) \big \Vert a_l^{-h_n}-a_l\big \Vert _{L^\infty ({\mathbb {R}})} \nonumber \\&\quad + c_{X({\mathbb {R}},w)}\Vert \psi \Vert _{L^1({\mathbb {R}})} \big \Vert (a_l^{-h_n})'-a_l'\big \Vert _{L^\infty ({\mathbb {R}})}. \end{aligned}$$

(6.19)

Combining (6.14)–(6.15) and (6.18)–(6.19) with (6.9)–(6.10), we see that equalities (6.12)–(6.13) hold for every $f\in {\mathcal {S}}_0({\mathbb {R}})$. Therefore, (6.11) are fulfilled for every $f\in X({\mathbb {R}},w)$. Hence, by the Banach-Steinhaus theorem (see, e.g., [34, Theorem 1.4.2]),

$$\begin{aligned} \Vert a_r\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}&= \Vert W^0(a_r)\Vert _{{\mathcal {B}}(X({\mathbb {R}},w))} \le \liminf _{n\rightarrow \infty }\Vert e_{h_n}W^0(a)e_{-h_n}I\Vert _{{\mathcal {B}}(X({\mathbb {R}},w))} \\&\le \Vert W^0(a)\Vert _{{\mathcal {B}}(X({\mathbb {R}},w))}=\Vert a\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}} \end{aligned}$$

and, analogously,

$$\begin{aligned} \Vert a_l\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}&= \Vert W^0(a_l)\Vert _{{\mathcal {B}}(X({\mathbb {R}},w))} \le \liminf _{n\rightarrow \infty }\Vert e_{-h_n}W^0(a)e_{h_n}I\Vert _{{\mathcal {B}}(X({\mathbb {R}},w))} \\&\le \Vert W^0(a)\Vert _{{\mathcal {B}}(X({\mathbb {R}},w))}=\Vert a\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}, \end{aligned}$$

which completes the proof of (6.7). $\square$

Now we extend the previous result for functions a of the form (6.8) with general $a_l,a_r\in AP_{X({\mathbb {R}},w)}$ and $a_0\in C_{0,X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}})$, keeping the nice function v as above.

Lemma 6.4

Let $X({\mathbb {R}})$ be a separable rearrangement-invariant Banach function space with the Boyd indices satisfying $0<\alpha _X$, $\beta _X<1$. Suppose that $w\in A_{1/\alpha _X}^0({\mathbb {R}})\cap A_{1/\beta _X}^0({\mathbb {R}})$. Let $v\in C({\overline{{\mathbb {R}}}})$ be any real-valued monotonically increasing function such that there exists a point $x_0>0$ such that $v(x)=0$ for $x<-x_0$ and $v(x)=1$ for $x>x_0$. If $a_l,a_r\in AP_{X({\mathbb {R}},w)}$, $a_0\in C_{0,X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}})$, where $C_{0,X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}})$ is defined by (3.17), and a is given by equality (6.8), then inequalities (6.7) hold.

Proof

By the definition of $AP_{X({\mathbb {R}},w)}$, there are sequences $\{a_l^{(n)}\}_{n\in {\mathbb {N}}},\{a_r^{(n)}\}_{n\in {\mathbb {N}}}$ in APP such that

$$\begin{aligned} \lim _{n\rightarrow \infty }\big \Vert a_l^{(n)}-a_l\big \Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}=0, \quad \lim _{n\rightarrow \infty }\big \Vert a_r^{(n)}-a_r\big \Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}=0. \end{aligned}$$

(6.20)

On the other hand, by Theorem 3.3, there is a sequence $\{a_0^{(n)}\}_{n\in {\mathbb {N}}}$ in $C_c^\infty ({\mathbb {R}})$ such that

$$\begin{aligned} \lim _{n\rightarrow \infty }\big \Vert a_0^{(n)}-a_0\big \Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}=0. \end{aligned}$$

(6.21)

For $n\in {\mathbb {N}}$, consider the functions

$$\begin{aligned} a^{(n)}:=(1-v)a_l^{(n)}+va_r^{(n)}+a_0^{(n)}. \end{aligned}$$

(6.22)

It follows from equalities (6.20)–(6.22) and Lemma 6.3 that

$$\begin{aligned} \Vert a_l\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}&= \lim _{n\rightarrow \infty }\big \Vert a_l^{(n)}\big \Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}} \le \lim _{n\rightarrow \infty }\big \Vert a^{(n)}\big \Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}} = \Vert a\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}, \\ \Vert a_r\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}&= \lim _{n\rightarrow \infty }\big \Vert a_r^{(n)}\big \Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}} \le \lim _{n\rightarrow \infty }\big \Vert a^{(n)}\big \Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}} = \Vert a\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}, \end{aligned}$$

which completes the proof of inequalities (6.7). $\square$

Now we observe that the algebra ${\mathcal {A}}_u$ does not depend on the particular choice of a real-valued monotonically increasing function $u\in C({\overline{{\mathbb {R}}}})$ satisfying conditions (6.1).

Lemma 6.5

Let $X({\mathbb {R}})$ be a separable rearrangement-invariant Banach function space with the Boyd indices satisfying $0<\alpha _X$, $\beta _X<1$. Suppose that $w\in A_{1/\alpha _X}^0({\mathbb {R}})\cap A_{1/\beta _X}^0({\mathbb {R}})$. Let $u,v\in C({\overline{{\mathbb {R}}}})$ be two real-valued monotonically increasing functions such that

$$\begin{aligned} u(-\infty )=v(-\infty )=0, \quad u(+\infty )=v(+\infty )=1. \end{aligned}$$

Then ${\mathcal {A}}_u={\mathcal {A}}_v$.

Proof

If $a\in {\mathcal {A}}_u$, then $a=(1-u)a_l+ua_r+a_0$ for some $a_l,a_r\in AP_{X({\mathbb {R}},w)}$ and $a_0\in C_{0,X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}})$. On the other hand, $a=(1-v)a_l+va_r+b_0$ with

$$\begin{aligned} b_0=(v-u)a_l+(u-v)a_r+a_0=(u-v)(a_r-a_l)+a_0. \end{aligned}$$

Since the functions u, v are monotonically increasing, we have $u,v\in V({\mathbb {R}})$. Hence $u-v\in V({\mathbb {R}})\cap C({\overline{{\mathbb {R}}}})$ and

$$\begin{aligned} u(+\infty )-v(+\infty )=u(-\infty )-v(-\infty )=0. \end{aligned}$$

Thus $u-v\in C({{\dot{{\mathbb {R}}}}})\cap V({\mathbb {R}})\subset C_{X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}})$ and $(u-v)(\infty )=0$. Since the function $a_r-a_l$ belongs to $AP_{X({\mathbb {R}},w)}$, it follows from Lemma 5.3 that

$$\begin{aligned} (u-v)(a_r-a_l)\in C_{0,X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}}). \end{aligned}$$

Then $b_0\in C_{0,X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}})$ and $a\in {\mathcal {A}}_v$. Therefore ${\mathcal {A}}_u\subset {\mathcal {A}}_v$. It can be shown analogously that ${\mathcal {A}}_v\subset {\mathcal {A}}_u$. Thus ${\mathcal {A}}_u={\mathcal {A}}_v$. $\square$

Combining Lemmas 6.4–6.5, we arrive at the main result of this subsection.

Theorem 6.6

Let $X({\mathbb {R}})$ be a separable rearrangement-invariant Banach function space with the Boyd indices satisfying $0<\alpha _X$, $\beta _X<1$. Suppose that $w\in A_{1/\alpha _X}^0({\mathbb {R}})\cap A_{1/\beta _X}^0({\mathbb {R}})$. Let $u\in C({\overline{{\mathbb {R}}}})$ be a real-valued monotonically increasing function such that $u(-\infty )=0$ and $u(+\infty )=1$. If $a\in {\mathcal {A}}_u$, that is,

$$\begin{aligned} a=(1-u)a_l+ua_r+a_0 \quad \text{ with }\quad a_l,a_r\in AP_{X({\mathbb {R}},w)},a_0\in C_{0,X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}}), \end{aligned}$$

where $C_{0,X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}})$ is defined by (3.17), then inequalities (6.7) hold.

6.3 Proof of Theorem 1.2

The idea of the proof is borrowed from the proof of [10, Theorem 1.21]. If $a\in AP_{X({\mathbb {R}},w)}$, then $a=(1-u)a+ua+0$, whence $a\in {\mathcal {A}}_u$. If $f\in C_{X({\mathbb {R}},w)}({\overline{{\mathbb {R}}}})$, then the function $f_0=f-(1-u)f(-\infty )-uf(+\infty )$ belongs to $C_{0,X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}})$. Therefore $f=(1-u)f(-\infty )+uf(+\infty )+f_0\in {\mathcal {A}}_u$. These observations imply that

$$\begin{aligned} SAP_{X({\mathbb {R}},w)}\subset {\text {clos}}_{{\mathcal {M}}_{X({\mathbb {R}},w)}}({\mathcal {A}}_u). \end{aligned}$$

(6.23)

On the other hand, it is obvious that

$$\begin{aligned} {\text {clos}}_{{\mathcal {M}}_{X({\mathbb {R}},w)}}({\mathcal {A}}_u)\subset SAP_{X({\mathbb {R}},w)}. \end{aligned}$$

(6.24)

Combining (6.23)–(6.24), we arrive at the equality

$$\begin{aligned} SAP_{X({\mathbb {R}},w)}={\text {clos}}_{{\mathcal {M}}_{X({\mathbb {R}},w)}}({\mathcal {A}}_u). \end{aligned}$$

(6.25)

By Theorem 6.6, for every $a=(1-u)a_r+ua_r+a_0\in {\mathcal {A}}_u$ with $a_l,a_r\in AP_{X({\mathbb {R}},w)}$ and $a_0\in C_{0,X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}})$, one has

$$\begin{aligned} \Vert a_r\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}\le \Vert a\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}, \quad \Vert a_l\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}\le \Vert a\Vert _{{\mathcal {M}}_{X({\mathbb {R}},w)}}. \end{aligned}$$

(6.26)

Consequently, if $\big \{(1-u)a_l^{(n)}+ua_r^{(n)}+a_0^{(n)}\big \}_{n\in {\mathbb {N}}}$ is a Cauchy sequence in ${\mathcal {A}}_u$, where $\big \{a_l^{(n)}\big \}_{n\in {\mathbb {N}}}, \big \{a_r^{(n)}\big \}_{n\in {\mathbb {N}}}$ are sequences in $AP_{X({\mathbb {R}},w)}$ and $\big \{a_0^{(n)}\big \}_{n\in {\mathbb {N}}}$ is a sequence in $C_{0,X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}})$, then $\big \{a_l^{(n)}\big \}_{n\in {\mathbb {N}}}$ and $\big \{a_r^{(n)}\big \}_{n\in {\mathbb {N}}}$ are Cauchy sequences in $AP_{X({\mathbb {R}},w)}$. Consequently, $\big \{a_0^{(n)}\big \}_{n\in {\mathbb {N}}}$ is a Cauchy sequence in $C_{0,X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}})$. Since $AP_{X({\mathbb {R}},w)}$ is closed by its definition and $C_{0,X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}})$ is closed in view of Theorem 3.3, we conclude that the limits

$$\begin{aligned} a_l:=\lim _{n\rightarrow \infty }a_l^{(n)}, \quad a_r=\lim _{n\rightarrow \infty }a_r^{(n)} \end{aligned}$$

belong to $AP_{X({\mathbb {R}},w)}$ and that the limit

$$\begin{aligned} a_0:=\lim _{n\rightarrow \infty }a_0^{(n)} \end{aligned}$$

belongs to $C_{0,X({\mathbb {R}},w)}({{\dot{{\mathbb {R}}}}})$. Therefore, the limit

$$\begin{aligned} \lim _{n\rightarrow \infty } \big ((1-u)a_l^{(n)}+ua_r^{(n)}+a_0^{(n)}\big ) \end{aligned}$$

belongs to ${\mathcal {A}}_u$. Thus

$$\begin{aligned} {\text {clos}}_{{\mathcal {M}}_{X({\mathbb {R}},w)}}({\mathcal {A}}_u)={\mathcal {A}}_u. \end{aligned}$$

(6.27)

It follows from (6.25) and (6.27) that ${\mathcal {A}}_u=SAP_{X({\mathbb {R}},w)}$. In particular, every function $a\in SAP_{X({\mathbb {R}},w)}$ is of the form

$$\begin{aligned} a=(1-u)a_l+ua_r+a_0 \end{aligned}$$

(6.28)

with $a_l,a_r\in AP_{X({\mathbb {R}},w)}$ and $a_0\in C_{0,X({\mathbb {R}},w)}$. We infer from (6.26) that the representation (6.28) is unique for the function u. Moreover, the proof of Lemma 6.5 shows that $a_l,a_r\in AP_{X({\mathbb {R}},w)}$ are independent of the particular choice of the function u. By Lemma 6.1, the mappings $a\mapsto a_l$ and $a\mapsto a_r$ are algebraic homomorphisms of ${\mathcal {A}}_u=SAP_{X({\mathbb {R}},w)}$ onto $AP_{X({\mathbb {R}},w)}$. In view of (6.26), these homomorphisms are Banach algebra homomorphisms of the Banach algebra $SAP_{X({\mathbb {R}},w)}$ onto the Banach algebra $AP_{X({\mathbb {R}},w)}$ and the norms of these homomorphisms are not greater than one. For every function $a\in AP_{X({\mathbb {R}},w)}$, we have equalities in (6.26) because

$$\begin{aligned} a=(1-u)a+ua+0=a_l=a_r. \end{aligned}$$

Thus, the norms of the homomorphisms $a\mapsto a_l$ and $a\mapsto a_l$ are equal to one. $\square$