Fourier Convolution Operators with Symbols Equivalent to Zero at Infinity on Banach Function Spaces

Abstract

We study Fourier convolution operators W ⁰(a) with symbols equivalent to zero at infinity on a separable Banach function space $X(\mathbb {R})$ such that the Hardy-Littlewood maximal operator is bounded on $X(\mathbb {R})$ and on its associate space $X'(\mathbb {R})$. We show that the limit operators of W ⁰(a) are all equal to zero.

1 Introduction

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

$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathrm{(A1)} & &\displaystyle \rho(f)=0 \Leftrightarrow f=0\ \mbox{a.e.}, \quad \rho(af)=a\rho(f), \quad \rho(f+g) \le \rho(f)+\rho(g),\\ \mathrm{(A2)} & &\displaystyle 0\le g \le f \ \mbox{a.e.} \ \Rightarrow \ \rho(g) \le \rho(f) \quad \mbox{(the lattice property)}, \\ \mathrm{(A3)} & &\displaystyle 0\le f_n \uparrow f \ \mbox{a.e.} \ \Rightarrow \ \rho(f_n) \uparrow \rho(f)\quad \mbox{(the Fatou property)},\\ \mathrm{(A4)} & &\displaystyle |E|<\infty \Rightarrow \rho(\chi_E) <\infty,\\ \mathrm{(A5)} & &\displaystyle |E|<\infty \Rightarrow \int_E f(x)\,dx \le C_E\rho(f) \end{array} \end{aligned} $$

with C _E ∈ (0, ∞) which may depend on E and ρ but is independent of f. When functions differing only on a set of measure zero are identified, the set $X(\mathbb {R})$ of all functions $f\in \mathfrak {M}(\mathbb {R})$ for which ρ(|f|) < ∞ is called a Banach function space. For each $f\in X(\mathbb {R})$, the norm of f is defined by $\left \|f\right \|{ }_{X(\mathbb {R})} :=\rho (|f|)$. Under the natural linear space operations and under this norm, the set $X(\mathbb {R})$ becomes a Banach space (see [1, Chap. 1, Theorems 1.4 and 1.6]). If ρ is a Banach function norm, its associate norm ρ′ is defined on $\mathfrak {M}^+(\mathbb {R})$ by

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

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 ρ′ is called the associate space (Köthe dual) of $X(\mathbb {R})$. The associate space $X'(\mathbb {R})$ is naturally identified with a subspace of the (Banach) dual space $[X(\mathbb {R})]^*$.

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

$$\displaystyle \begin{aligned} (\mathcal{F} f)(x):=\widehat{f}(x):=\int_{\mathbb{R}} f(t)e^{itx}\,dt, \quad x\in\mathbb{R}, \end{aligned}$$

and let $\mathcal {F}^{-1}:L^2(\mathbb {R})\to L^2(\mathbb {R})$ be the inverse of $\mathcal {F}$. It is well known that the Fourier convolution operator

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

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. Then by Karlovich and Spitkovsky [9, Lemma 2.12(a)], $L^2(\mathbb {R})\cap X(\mathbb {R})$ is dense in $X(\mathbb {R})$. A function $a\in L^\infty (\mathbb {R})$ is called a Fourier multiplier on $X(\mathbb {R})$ if the convolution operator W ⁰(a) maps $L^2(\mathbb {R})\cap X(\mathbb {R})$ into $X(\mathbb {R})$ 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

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

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

Recall that the (non-centered) Hardy-Littlewood maximal operator M of a function $f\in L_{\mathrm {loc}}^1(\mathbb {R})$ is defined by

$$\displaystyle \begin{aligned} (M f)(x):=\sup_{J\ni x}\frac{1}{|J|}\int_J|f(y)|\,dy, \end{aligned}$$

where the supremum is taken over all finite intervals $J\subset \mathbb {R}$ containing x.

Let $V(\mathbb {R})$ be the Banach algebra of all functions $a:\mathbb {R}\to \mathbb {C}$ with finite total variation

$$\displaystyle \begin{aligned} V(a):=\sup\sum_{i=1}^n|a(t_i)-a(t_{i-1})|, \end{aligned}$$

where the supremum is taken over all partitions −∞ < t ₀ < ⋯ < t _n < +∞ of the real line $\mathbb {R}$ and the norm in $V(\mathbb {R})$ is given by $\|a\|{ }_{V}:=\|a\|{ }_{L^\infty (\mathbb {R})}+V(a)$.

Theorem 1.1

Let $X(\mathbb {R})$ be a separable Banach function space such that the Hardy-Littlewood maximal operator M is bounded on $X(\mathbb {R})$ and on its associate space $X'(\mathbb {R})$ . If $a\in V(\mathbb {R})$ , then the convolution operator W ⁰(a) is bounded on the space $X(\mathbb {R})$ and

$$\displaystyle \begin{aligned} \|W^0(a)\|{}_{\mathcal{B}(X(\mathbb{R}))} \le c_{X}\|a\|{}_V \end{aligned} $$

(1)

where c _X is a positive constant depending only on $X(\mathbb {R})$.

This result follows from [5, Theorem 4.3]. Inequality (1) is usually called the Stechkin type inequality (see also [6, inequality (2.4)]).

Following [3, p. 140], two Fourier multipliers $c,d\in \mathcal {M}_{X(\mathbb {R})}$ are called equivalent at infinity if

$$\displaystyle \begin{aligned} \lim_{N\to\infty}\left\|\chi_{\mathbb{R}\setminus[-N,N]}(c-d)\right\|{}_{\mathcal{M}_{X(\mathbb{R})}}=0. \end{aligned}$$

In the latter case we will write .

The aim of this paper is to start the study of Fourier convolution operators with symbols equivalent at infinity to well behaved symbols by the method of limit operators in the context of Banach function spaces. We refer to [10] for a general theory of limit operators and to [6,7,8] for its applications to the study of Fourier convolution operators with piecewise slowly oscillating symbols on Lebesgue spaces with Muckenhoupt weights, constituting a remarkable example of Banach function spaces.

For a sequence of operators $\{A_n\}_{n\in \mathbb {N}}\subset \mathcal {B}(X(\mathbb {R}))$, let denote the strong limit of the sequence, if it exists. For $\lambda ,x\in \mathbb {R}$, consider the function e _λ(x) := e ^iλx. Let $T\in \mathcal {B}(X(\mathbb {R}))$ and $h=\{h_n\}_{n\in \mathbb {N}}\subset (0,\infty )$ be a sequence satisfying h _n → +∞ as n →∞. The strong limit

is called the limit operator of T related to the sequence $h=\{h_n\}_{n\in \mathbb {N}}$, if it exists.

Theorem 1.2 (Main Result)

Let $X(\mathbb {R})$ be a separable Banach function space such that the Hardy-Littlewood maximal operator M is bounded on the space $X(\mathbb {R})$ and on its associate space $X'(\mathbb {R})$ . If $a\in \mathcal {M}_{X(\mathbb {R})}$ is such that , then for every sequence $h=\{h_n\}_{n\in \mathbb {N}}\subset (0,\infty )$ , satisfying h _n → +∞ as n →∞, the limit operator of W ⁰(a) related to the sequence h is the zero operator.

As usual, let $C_0^\infty (\mathbb {R})$ denote the set of all infinitely differentiable functions with compact support and let $\mathcal {S}(\mathbb {R})$ be the Schwartz space of rapidly decreasing smooth functions. Finally, 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.

The paper is organized as follows. In Sect. 2, we discuss approximation by mollifiers in separable Banach function spaces such that M is bounded on $X(\mathbb {R})$. In Sect. 3, we show that under the assumptions of the previous section, the set $\mathcal {S}_0(\mathbb {R})$ is dense in the space $X(\mathbb {R})$. Finally, in Sect. 4, we prove Theorem 1.2, essentially using the density of $\mathcal {S}_0(\mathbb {R})$ in the space $X(\mathbb {R})$.

2 Mollification in Separable Banach Function Spaces

The following auxiliary statement might be of independent interest.

Theorem 2.1

Let $\varphi \in L^1(\mathbb {R})$ satisfy $\int _{\mathbb {R}}\varphi (x)\,dx=1$ and

$$\displaystyle \begin{aligned} \varphi_\delta(x):=\delta^{-1}\varphi(x/\delta),\quad x\in\mathbb{R},\quad \delta>0. \end{aligned} $$

(2)

Suppose that the radial majorant of φ given by $\Phi (x):=\sup \limits _{|y|\ge |x|}|\varphi (y)|$ belongs to $L^1(\mathbb {R})$ . If $X(\mathbb {R})$ is a Banach function space such that the Hardy-Littlewood maximal operator M is bounded on the space $X(\mathbb {R})$ , then for all $f\in X(\mathbb {R})$,

$$\displaystyle \begin{aligned} \sup_{\delta>0}\|f*\varphi_\delta\|{}_{X(\mathbb{R})} \le L\|f\|{}_{X(\mathbb{R})}, \end{aligned} $$

(3)

where $L:=\|\Phi \|{ }_{L^1(\mathbb {R})}\|M\|{ }_{\mathcal {B}(X(\mathbb {R}))}$ and $\|M\|{ }_{\mathcal {B}(X(\mathbb {R}))}$ denotes the norm of the sublinear operator M on the space $X(\mathbb {R})$ . If, in addition, the space $X(\mathbb {R})$ is separable, then for all $f\in X(\mathbb {R})$,

$$\displaystyle \begin{aligned} \lim_{\delta\to 0^+} \|f*\varphi_\delta-f\|{}_{X(\mathbb{R})}=0. \end{aligned} $$

(4)

Proof

The idea of the proof is borrowed from [11, Theorem 2.4]. By the proof of [2, Lemma 5.7], for every $f\in L_{\mathrm {loc}}^1(\mathbb {R})$,

$$\displaystyle \begin{aligned} \sup_{\delta>0}|(f*\varphi_\delta)(x)|\le \|\Phi\|{}_{L^1(\mathbb{R})}(M f)(x), \quad x\in\mathbb{R}. \end{aligned} $$

(5)

Inequality (3) follows from inequality (5), the boundedness of the Hardy-Littlewood maximal operator M on the space $X(\mathbb {R})$ and Axiom (A2).

Now assume that the space $X(\mathbb {R})$ is separable. Then by Karlovich and Spitkovsky [9, Lemma 2.12(a)], the set $C_0^\infty (\mathbb {R})$ is dense in the space $X(\mathbb {R})$. Take $f\in X(\mathbb {R})$ and fix ε > 0. Then there exists $g\in C_0^\infty (\mathbb {R})$ such that

$$\displaystyle \begin{aligned} \|f-g\|{}_{X(\mathbb{R})}<\frac{\varepsilon}{2(L+1)}. \end{aligned} $$

(6)

Hence for all δ > 0,

$$\displaystyle \begin{aligned} \|f*\varphi_\delta-f\|{}_{X(\mathbb{R})} \le \|(f-g)*\varphi_\delta-(f-g)\|{}_{X(\mathbb{R})} + \|g*\varphi_\delta-g\|{}_{X(\mathbb{R})}. \end{aligned} $$

(7)

Taking into account inequalities (3) and (6), we obtain for all δ > 0,

$$\displaystyle \begin{aligned} \|(f-g)*\varphi_\delta-(f-g)\|{}_{X(\mathbb{R})} &\le \|(f-g)*\varphi_\delta\|{}_{X(\mathbb{R})}+\|f-g\|{}_{X(\mathbb{R})} \\ &\le (L+1)\|f-g\|{}_{X(\mathbb{R})}<\varepsilon/2. {} \end{aligned} $$

(8)

Let {δ _n} be an arbitrary sequence of positive numbers such that δ _n → 0 as n →∞. Since $g\in C_0^\infty (\mathbb {R})$, it follows from [13, Chap. III, Theorem 2(b)] that

$$\displaystyle \begin{aligned} \lim_{n\to\infty}(g*\varphi_{\delta_n})(x)=g(x) \quad \mbox{for a.e.}\quad x\in\mathbb{R}. \end{aligned} $$

(9)

In view of (5), we have for all $n\in \mathbb {N}$,

$$\displaystyle \begin{aligned} |(g*\varphi_{\delta_n})(x)|\le \|\Phi\|{}_{L^1(\mathbb{R})}(M g)(x),\quad x\in\mathbb{R}. \end{aligned} $$

(10)

Since $g\in C_0^\infty (\mathbb {R})\subset X(\mathbb {R})$ and the Hardy-Littlewood maximal operator M is bounded on the space $X(\mathbb {R})$, we see that $M g\in X(\mathbb {R})$. Then Mg has absolutely continuous norm because the Banach function space $X(\mathbb {R})$ is separable (see [1, Chap. 1, Definition 3.1 and Corollary 5.6]). It follows from (9)–(10) and the dominated convergence theorem for Banach function spaces (see [1, Chap. 1, Proposition 3.6]) that

$$\displaystyle \begin{aligned} \lim_{n\to\infty}\|g*\varphi_{\delta_n}-g\|{}_{X(\mathbb{R})}=0. \end{aligned}$$

Since the sequence {δ _n} is arbitrary, this means that one can find δ ₀ > 0 such that for all δ ∈ (0, δ ₀),

$$\displaystyle \begin{aligned} \|g*\varphi_\delta-g\|{}_{X(\mathbb{R})}<\varepsilon/2. \end{aligned} $$

(11)

Combining (7), (8), and (11), we see that for all δ ∈ (0, δ ₀) one has

$$\displaystyle \begin{aligned} \|f*\varphi_\delta-f\|{}_{X(\mathbb{R})}<\varepsilon, \end{aligned}$$

which immediately implies (4). □

3 Density of the Set $\mathcal {S}_0(\mathbb {R})$

Lemma 3.1

Let $X(\mathbb {R})$ be a Banach function space such that the Hardy-Littlewood maximal operator M is bounded on $X(\mathbb {R})$ . Then $\mathcal {S}(\mathbb {R})\subset X(\mathbb {R})$.

Proof

Suppose that $f\in \mathcal {S}(\mathbb {R})$. Then, in particular,

$$\displaystyle \begin{aligned} \rho_0(f):=\sup_{x\in\mathbb{R}}|f(x)|<\infty, \quad \rho_1(f):=\sup_{x\in\mathbb{R}}|xf(x)|<\infty. \end{aligned}$$

By Grafakos [4, Example 2.1.4],

$$\displaystyle \begin{aligned} \frac{\chi_{\mathbb{R}\setminus[-1,1]}(x)}{|x|} \le \chi_{\mathbb{R}\setminus[-1,1]}(x)(M\chi_{[-1,1]})(x). \end{aligned} $$

(12)

Since the function χ _[−1,1] belongs to $X(\mathbb {R})$ by Axiom (A4) and since the operator M is bounded on the space $X(\mathbb {R})$, we have $M\chi _{[-1,1]}\in X(\mathbb {R})$. Let ψ(x) = |x|. Then in view of (12) and Axiom (A2), we obtain

$$\displaystyle \begin{aligned} \|f\|{}_{X(\mathbb{R})} &\le \left\|\chi_{[-1,1]}f\right\|{}_{X(\mathbb{R})} + \left\|\chi_{\mathbb{R}\setminus[-1,1]}\psi f M\chi_{[-1,1]}\right\|{}_{X(\mathbb{R})} \\ &\le \rho_0(f)\left\|\chi_{[-1,1]}\right\|{}_{X(\mathbb{R})} + \rho_1(f)\left\|M\chi_{[-1,1]}\right\|{}_{X(\mathbb{R})}. \end{aligned} $$

Thus, $f\in X(\mathbb {R})$. □

Theorem 3.2

Let $X(\mathbb {R})$ be a separable Banach function space such that the Hardy-Littlewood maximal operator M is bounded on $X(\mathbb {R})$ . Then the set $\mathcal {S}_0(\mathbb {R})$ is dense in the space $X(\mathbb {R})$.

Proof

Let $f\in X(\mathbb {R})$. Fix ε > 0. By Karlovich and Spitkovsky [9, Lemma 2.12(a)], there exists a function $g\in C_0^\infty (\mathbb {R})$ such that

$$\displaystyle \begin{aligned} \|f-g\|{}_{X(\mathbb{R})}<\varepsilon/2. \end{aligned} $$

(13)

Let

$$\displaystyle \begin{aligned} { \varrho(x):=\left\{\begin{array}{lll} e^{1/(x^2-1)} &\mbox{if}& |x|<1, \\ 0 &\mbox{if}& |x|\ge 1, \end{array}\right. \quad \varphi(x):=\frac{(\mathcal{F}^{-1}\varrho)(x)}{\int_{\mathbb{R}}(\mathcal{F}^{-1}\varrho)(y)\,dy}, \quad x\in\mathbb{R}. } \end{aligned}$$

As $\varrho \in C_0^\infty (\mathbb {R})\subset \mathcal {S}(\mathbb {R})$, it follows immediately from [4, Corollary 2.2.15] that $\varphi \in \mathcal {S}_0(\mathbb {R})$. For all δ > 0, we define the family of functions φ _δ by (2). Since $g\in C_0^\infty (\mathbb {R})$ and $\varphi _\delta \in \mathcal {S}(\mathbb {R})$, we infer from [4, Proposition 2.2.11(12)] that

$$\displaystyle \begin{aligned}{}[\mathcal{F}(g*\varphi_\delta)](x) = (\mathcal{F} g)(x)(\mathcal{F}\varphi_\delta)(x) = (\mathcal{F} g)(x)(\mathcal{F}\varphi)(\delta x), \quad x\in\mathbb{R}. \end{aligned}$$

As $\mathcal {F}\varphi$ has compact support, we conclude that $\mathcal {F}(g*\varphi _\delta )$ also has compact support. Thus $g*\varphi _\delta \in \mathcal {S}_0(\mathbb {R})$ for every δ > 0. By Lemma 3.1, $g*\varphi _\delta \in X(\mathbb {R})$.

By the definition of the Schwartz class $\mathcal {S}(\mathbb {R})$, there are constants C _n > 0 such that

$$\displaystyle \begin{aligned} |\varphi(x)|\le C_n(1+|x|)^{-n}, \quad x\in\mathbb{R}, \quad n\in\mathbb{N}\cup\{0\}. \end{aligned}$$

Then

$$\displaystyle \begin{aligned} \Phi(x)=\sup_{|y|\ge|x|}|\varphi(y)|\le C_n\sup_{|y|\ge |x|}(1+|y|)^{-n} =C_n(1+|x|)^{-n} \end{aligned}$$

for $x\in \mathbb {R}$ and $n\in \mathbb {N}\cup \{0\}$. This estimate implies that the radial majorant Φ of the function φ is integrable.

Since $\Phi \in L^1(\mathbb {R})$, the space $X(\mathbb {R})$ is separable, and the Hardy-Littlewood maximal operator M is bounded on $X(\mathbb {R})$, it follows from Theorem 2.1 that there is a δ > 0 such that

$$\displaystyle \begin{aligned} \|g*\varphi_\delta-g\|{}_{X(\mathbb{R})}<\varepsilon/2. \end{aligned} $$

(14)

Combining (13) and (14), we see that for every ε > 0 there is a δ > 0 such that $\|f-g*\varphi _\delta \|{ }_{X(\mathbb {R})}<\varepsilon$. Since $g*\varphi _\delta \in \mathcal {S}_0(\mathbb {R})$, the proof is completed. □

4 Proof of Theorem 1.2

Fix a sequence $\{h_n\}_{n\in \mathbb {N}}\subset (0,\infty )$ such that h _n → +∞ as n →∞. For every function $f\in \mathcal {S}_0(\mathbb {R})$ there exists a segment $K=[x_1,x_2]\subset \mathbb {R}$ such that $\operatorname {supp}\mathcal {F} f\subset [x_1,x_2]$. Therefore

$$\displaystyle \begin{aligned} e_{h_n}W^0(a)e_{h_n}^{-1}f &= W^0[a(\cdot+h_n)]f=\mathcal{F}^{-1}[a(\cdot+h_n)\chi_K]\mathcal{F} f \\ &= W^0(a\chi_{K+h_n})f, {} \end{aligned} $$

(15)

where K + h _n = {x + h _n : x ∈ K}.

Fix ε > 0. Without loss of generality we may assume that f≠0. As , there exists N > 0 such that

$$\displaystyle \begin{aligned} \left\|\chi_{\mathbb{R}\setminus[-N,N]}a\right\|{}_{\mathcal{M}_{X(\mathbb{R})}} < \frac{\varepsilon}{3c_X\|f\|{}_{X(\mathbb{R})}}, \end{aligned} $$

(16)

where c _X > 0 is the constant from Stechkin’s type inequality (1). Since h _n → +∞ as n →∞, we conclude that there exists $n_0\in \mathbb {N}$ such that for all n > n ₀, one has $K+h_n\subset (N,+\infty )\subset \mathbb {R}\setminus [-N,N]$. Therefore, for n > n ₀, we have

$$\displaystyle \begin{aligned} a\chi_{K+h_n}=\chi_{\mathbb{R}\setminus[-N,N]}a\chi_{K+h_n}. \end{aligned} $$

(17)

By Theorem 1.1, for every n > n ₀, we have

$$\displaystyle \begin{aligned} \left\|\chi_{K+h_n}\right\|{}_{\mathcal{M}_{X(\mathbb{R})}} \le c_X\left\|\chi_{K+h_n}\right\|{}_{V}= 3c_X. \end{aligned} $$

(18)

Combining (15)– (18), we see that for n > n ₀,

$$\displaystyle \begin{aligned} \left\|e_{h_n}W^0(a)e_{h_n}^{-1}f\right\|{}_{X(\mathbb{R})} &\le \left\|\chi_{R\setminus[-N,N]}a\chi_{K+h_n}\right\|{}_{\mathcal{M}_{X(\mathbb{R})}}\|f\|{}_{X(\mathbb{R})} \\ &\le \left\|\chi_{R\setminus[-N,N]}a\right\|{}_{\mathcal{M}_{X(\mathbb{R})}} \left\|\chi_{K+h_n}\right\|{}_{\mathcal{M}_{X(\mathbb{R})}}\|f\|{}_{X(\mathbb{R})} <\varepsilon. \end{aligned} $$

Hence, for every $f\in \mathcal {S}_0(\mathbb {R})$,

$$\displaystyle \begin{aligned} \lim_{n\to\infty}\left\|e_{h_n}W^0(a)e_{h_n}^{-1}f\right\|{}_{X(\mathbb{R})}=0. \end{aligned}$$

Since $\mathcal {S}_0(\mathbb {R})$ is dense in $X(\mathbb {R})$ (see Theorem 3.2), the latter equality immediately implies that

on the space $X(\mathbb {R})$ in view of [12, Lemma 1.4.1(ii)]. □