Interpolation between $H^{p(\cdot)}(\mathbb R^n)$ and $L^\infty(\mathbb R^n)$: Real Method

TL;DR

This paper establishes a real interpolation theorem between variable Hardy spaces and $L^ abla$ spaces, characterizing the structure of variable weak Hardy spaces and their relation to variable Lebesgue spaces.

Contribution

It introduces a decomposition for variable weak Hardy space distributions and proves a new interpolation theorem linking variable Hardy and $L^ abla$ spaces.

Findings

Interpolation formula for $(H^{p( abla)},L^ abla)$ spaces.

Equivalence of variable weak Hardy and Lebesgue spaces under certain conditions.

Characterization of variable weak Hardy spaces as variable Lebesgue spaces.

Abstract

Let $p(\cdot):\ \mathbb R^n\to(0,\infty)$ be a variable exponent function satisfying the globally log-H\"older continuous condition. In this article, the authors first obtain a decomposition for any distribution of the variable weak Hardy space into "good" and "bad" parts and then prove the following real interpolation theorem between the variable Hardy space $H^{p(\cdot)}(\mathbb R^n)$ and the space $L^{\infty}(\mathbb R^n)$: \begin{equation*} (H^{p(\cdot)}(\mathbb R^n),L^{\infty}(\mathbb R^n))_{\theta,\infty} =W\!H^{p(\cdot)/(1-\theta)}(\mathbb R^n),\quad \theta\in(0,1), \end{equation*} where $W\!H^{p(\cdot)/(1-\theta)}(\mathbb R^n)$ denotes the variable weak Hardy space. As an application, the variable weak Hardy space $W\!H^{p(\cdot)}(\mathbb R^n)$ with $p_-:=\mathop\mathrm{ess\,inf}_{x\in\rn}p(x)\in(1,\infty)$ is proved to coincide with the variable Lebesgue space $W\!L^{p(\cdot)}(\mathbb R^n)$.

Equations

$$(H^{p(\cdot )}({\mathbb{R}}^n),L^{\infty }({\mathbb{R}}^n){)}_{\theta ,\infty }=W{H}^{p(\cdot )/(1-\theta )}({\mathbb{R}}^n),\theta \in (0,1),$$

$$p_{-}:=\mathrm{missing}{essinf}_{x\in {\mathbb{R}}^n}p(x),$$

$$W{H}^p({\mathbb{R}}^n)=W{L}^p({\mathbb{R}}^n)$$

$$X_0+X_1:=\left\{f_0+f_1:f_0\in X_0\mathrm{and}{f}_1\in X_1\right\}.$$

$$K(t,f;X_0,X_1):=\inf \{\Vert f_0{\Vert }_{X_0}+t\Vert f_1{\Vert }_{X_1}:f=f_0+f_1,f_0\in X_0\mathrm{and}{f}_1\in X_1\}.$$

$$(X_0,X_1{)}_{\theta ,q}:=\left\{f\in X_0+X_1:\Vert f{\Vert }_{\theta ,q}<\infty \right\},$$

$$\Vert f{\Vert }_{\theta ,q}:=\{\begin{array}{ll}{\left[{\int }_0^{\infty }{\left\{t^{-\theta }K(t,f;X_0,X_1)\right\}}^q\frac{dt}{t}\right]}^{1/q}& \mathrm{if}q\in (0,\infty ),\\ \underset{t\in (0,\infty )}{\sup }{t}^{-\theta }K(t,f;X_0,X_1)& \mathrm{if}q=\infty .\end{array}$$

$$0<p_{-}\le p_{+}:=\mathrm{missing}{esssup}_{x\in {\mathbb{R}}^n}p(x)<\infty ,$$

$${\varrho }_{p(\cdot )}(f):={\int }_{{\mathbb{R}}^n}|f(x){|}^{p(x)}dx$$

$$\Vert f{\Vert }_{L^{p(\cdot )}({\mathbb{R}}^n)}:=\inf \left\{\lambda \in (0,\infty ):{\varrho }_{p(\cdot )}(f/\lambda )\le 1\right\}.$$

$$\Vert f{\Vert }_{W{L}^{p(\cdot )}({\mathbb{R}}^n)}:=\underset{\alpha \in (0,\infty )}{\sup }\alpha {\Vert {\chi }_{\{x\in {\mathbb{R}}^n:|f(x)|>\alpha \}}\Vert }_{L^{p(\cdot )}({\mathbb{R}}^n)}<\infty .$$

$$\Vert f+g{\Vert }_{L^{p(\cdot )}({\mathbb{R}}^n)}^{\underline{p}}\le \Vert f{\Vert }_{L^{p(\cdot )}({\mathbb{R}}^n)}^{\underline{p}}+\Vert g{\Vert }_{L^{p(\cdot )}({\mathbb{R}}^n)}^{\underline{p}},$$

$$\underline{p}:=\min \{p_{-},1\}$$

$$\Vert f+g{\Vert }_{W{L}^{p(\cdot )}({\mathbb{R}}^n)}^{\underline{p}}\le 2^{\underline{p}}\left[\Vert f{\Vert }_{W{L}^{p(\cdot )}({\mathbb{R}}^n)}^{\underline{p}}+\Vert g{\Vert }_{W{L}^{p(\cdot )}({\mathbb{R}}^n)}^{\underline{p}}\right].$$

$$|p(x)-p(y)|\le \frac{C_{\log }(p)}{\log (e+1/|x-y|)}$$

$$|p(x)-p_{\infty }|\le \frac{C_{\infty }}{\log (e+|x|)}.$$

$${\mathcal{F}}_N({\mathbb{R}}^n):=\left\{\psi \in \mathcal{S}({\mathbb{R}}^n):\sum _{\beta \in {\mathbb{Z}}_{+}^n,|\beta |\le N}\underset{x\in {\mathbb{R}}^n}{\sup }\left[(1+|x|{)}^N|D^{\beta }\psi (x)|\right]\le 1\right\},$$

$$f_{N,+}^{\ast }(x):=\sup \left\{|f\ast {\psi }_t(x)|:t\in (0,\infty )\mathrm{and}\psi \in {\mathcal{F}}_N({\mathbb{R}}^n)\right\},$$

$$f_{N,▽,a}^{\ast }(x):=\underset{\psi \in {\mathcal{F}}_N({\mathbb{R}}^n)}{\sup }\sup \left\{|f\ast {\psi }_t(y)|:y\in {\mathbb{R}}^n,t\in (0,\infty ),|y-x|<at\right\}.$$

$$\Vert f{\Vert }_{H^{p(\cdot )}({\mathbb{R}}^n)}:=\Vert f_{N,+}^{\ast }{\Vert }_{L^{p(\cdot )}({\mathbb{R}}^n)}.$$

$$\Vert f{\Vert }_{W{H}^{p(\cdot )}({\mathbb{R}}^n)}:=\Vert f_{N,+}^{\ast }{\Vert }_{W{L}^{p(\cdot )}({\mathbb{R}}^n)}.$$

$$(H^{p(\cdot )}({\mathbb{R}}^n),L^{\infty }({\mathbb{R}}^n){)}_{\theta ,\infty }=W{H}^{\tilde{p}(\cdot )}({\mathbb{R}}^n),$$

$$(H^p({\mathbb{R}}^n),L^{\infty }({\mathbb{R}}^n){)}_{\theta ,\infty }=W{H}^{p/(1-\theta )}({\mathbb{R}}^n),\theta \in (0,1).$$

$$(H^1({\mathbb{R}}^n),L^{\infty }({\mathbb{R}}^n){)}_{\theta ,\infty }=W{H}^{1/(1-\theta )}({\mathbb{R}}^n)=W{L}^{1/(1-\theta )}({\mathbb{R}}^n),\theta \in (0,1),$$

$$(L^p({\mathbb{R}}^n),L^{\infty }({\mathbb{R}}^n){)}_{\theta ,\infty }=W{L}^{p/(1-\theta )}({\mathbb{R}}^n),\theta \in (0,1).$$

$$W{H}^{\tilde{p}(\cdot )}({\mathbb{R}}^n)\subset (H^{p(\cdot )}({\mathbb{R}}^n),L^{\infty }({\mathbb{R}}^n){)}_{\theta ,\infty },$$

$$\left[W{H}^{\tilde{p}}({\mathbb{R}}^n)\cap \mathcal{S}({\mathbb{R}}^n)\right]\subset (H^p({\mathbb{R}}^n),L^{\infty }({\mathbb{R}}^n){)}_{\theta ,\infty }$$

$$s_0:=\lfloor n(1/p_{-}-1)\rfloor \mathrm{with}{p}_{-}\mathrm{as}\mathrm{in}\text{eqref}2.1x.$$

$${\int }_0^{\infty }\hat{\psi }(tx)\hat{\varphi }(tx)\frac{dt}{t}=1;$$

$$\hat{\eta }(x):={\int }_1^{\infty }\hat{\psi }(tx)\hat{\varphi }(tx)\frac{dt}{t}$$

Full text

Interpolation between $H^{p(\cdot)}(\mathbb{R}^{n})$

and $L^{\infty}(\mathbb{R}^{n})$: Real Method 00footnotetext: 2010 Mathematics Subject Classification. Primary 42B30; Secondary 42B35, 46B70. Key words and phrases. (weak) Hardy space, (weak) Lebesgue space, variable exponent, real interpolation. The first author is supported by the Construct Program of the Key Discipline in Hunan Province. This project is also supported by the National Natural Science Foundation of China (Grant Nos. 11571039, 11671185 and 11471042).

Ciqiang Zhuo, Dachun Yang 111Corresponding author / January 1, 2017.  and Wen Yuan

Abstract Let $p(\cdot):\ \mathbb{R}^{n}\to(0,\infty)$ be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the authors first obtain a decomposition for any distribution of the variable weak Hardy space into “good” and “bad” parts and then prove the following real interpolation theorem between the variable Hardy space $H^{p(\cdot)}(\mathbb{R}^{n})$ and the space $L^{\infty}(\mathbb{R}^{n})$:

[TABLE]

where $W\!H^{p(\cdot)/(1-\theta)}(\mathbb{R}^{n})$ denotes the variable weak Hardy space. As an application, the variable weak Hardy space $W\!H^{p(\cdot)}(\mathbb{R}^{n})$ with $p_{-}:=\mathop{\mathrm{missing}}{ess\,inf}_{x\in{{{\mathbb{R}}}^{n}}}p(x)\in(1,\infty)$ is proved to coincide with the variable Lebesgue space $W\!L^{p(\cdot)}(\mathbb{R}^{n})$.

1 Introduction

In recent years, theories of several variable function spaces, based on the variable Lebesgue space, have been rapidly developed (see, for example, [3, 4, 10, 13, 24, 30, 31, 34, 35]). Recall that the variable Lebesgue space ${L^{p(\cdot)}({{{\mathbb{R}}}^{n}})}$, with a variable exponent function $p(\cdot):\ {{{\mathbb{R}}}^{n}}\to(0,\infty)$, is a generalization of the classical Lebesgue space $L^{p}({{{\mathbb{R}}}^{n}})$. The study of variable Lebesgue spaces can be traced back to Orlicz [25], moreover, they have been the subject of more intensive study since the early work [22] of Kováčik and Rákosník and [14] of Fan and Zhao as well as [7] of Cruz-Uribe and [11] of Diening, because of their intrinsic interest for applications into harmonic analysis, partial differential equations and variational integrals with nonstandard growth conditions (see also [1, 2, 20, 32] and their references).

As a generalization of the classical Hardy space $H^{p}({{{\mathbb{R}}}^{n}})$ and the variable Lebesgue space ${L^{p(\cdot)}({{{\mathbb{R}}}^{n}})}$, the variable Hardy spaces ${H^{p(\cdot)}({{{\mathbb{R}}}^{n}})}$ were first investigated by Nakai and Sawano [24] with $p(\cdot)$ satisfying the globally log-Hölder continuous condition. In [24], they established the atomic characterizations of ${H^{p(\cdot)}({{{\mathbb{R}}}^{n}})}$, which were further applied to consider their dual spaces and the boundedness of singular integral operators on ${H^{p(\cdot)}({{{\mathbb{R}}}^{n}})}$. Later, Sawano [28] extended and improved the atomic characterization of ${H^{p(\cdot)}({{{\mathbb{R}}}^{n}})}$ in [24] and Zhuo et al. [36] gave their equivalent characterizations via (intrinsic) square functions including the (intrinsic) Lusin-area function, the (intrinsic) Littlewood-Paley $g$-function or $g_{\lambda}^{\ast}$-function. Independently, Cruz-Uribe and Wang [10] also studied the variable Hardy spaces ${H^{p(\cdot)}({{{\mathbb{R}}}^{n}})}$ with $p(\cdot)$ satisfying some conditions slightly weaker than those used in [24], and established their equivalent characterizations by means of radial or non-tangential maximal functions or atoms. However, the atomic characterization of ${H^{p(\cdot)}({{{\mathbb{R}}}^{n}})}$ obtained in [10] is very different from the classical case, which, in spirit, is closer to the atomic characterization for weighted Hardy spaces due to Strömberg and Torchinsky [29]. In addition, the characterizations of ${H^{p(\cdot)}({{{\mathbb{R}}}^{n}})}$ via Riesz transforms with $p(\cdot)$ satisfying the same conditions as in [10] were presented in [33].

Very recently, motivated by the well-known fact that, when studying the boundedness of some singular integral operators in the critical case, the weak Hardy space $W\!H^{p}({{{\mathbb{R}}}^{n}})$, with any $p\in(0,1]$, naturally appears as a proper substitute of the Hardy space $H^{p}({{{\mathbb{R}}}^{n}})$ (see [16, 23]), Yan et al. [31] introduced the variable weak Hardy space ${W\!H^{p(\cdot)}({{{\mathbb{R}}}^{n}})}$ and characterized these spaces via the radial or the non-tangential maximal functions, atoms, molecules, the Lusin-area function, the Littlewood-Paley $g$-function or $g_{\lambda}^{\ast}$-function. As an application, the authors in [31] established the boundedness of some convolutional $\delta$-type and non-convolutional $\gamma$-order Calderón-Zygmund operators from ${H^{p(\cdot)}({{{\mathbb{R}}}^{n}})}$ to ${W\!H^{p(\cdot)}({{{\mathbb{R}}}^{n}})}$ including the critical case when $p_{-}=\frac{n}{n+\delta}$ or $p_{-}=\frac{n}{n+\gamma}$, where

[TABLE]

which implies that the space ${W\!H^{p(\cdot)}({{{\mathbb{R}}}^{n}})}$ is a suitable substitute of the space ${H^{p(\cdot)}({{{\mathbb{R}}}^{n}})}$ in the study of boundedness of some singular integral operators in the critical case on ${H^{p(\cdot)}({{{\mathbb{R}}}^{n}})}$.

As was well known, Fefferman et al. [15] found that the weak Hardy space $W\!H^{p}({{{\mathbb{R}}}^{n}})$ naturally appears as the intermediate space of the classical Hardy space $H^{p}({{{\mathbb{R}}}^{n}})$ and the space $L^{\infty}({{{\mathbb{R}}}^{n}})$ under the real interpolation, which is another main motivation to develop the real-variable theory of $W\!H^{p}({{{\mathbb{R}}}^{n}})$. Therefore, it is natural and interesting to ask whether or not the variable weak Hardy space serves as the intermediate space between the variable Hardy space ${H^{p(\cdot)}({{{\mathbb{R}}}^{n}})}$ and the space $L^{\infty}({{{\mathbb{R}}}^{n}})$ via the real interpolation.

On the other hand, it is well known that, when $p\in(1,\infty)$,

[TABLE]

with equivalent quasi-norms (see [15]), where $W\!L^{p}({{{\mathbb{R}}}^{n}})$ denotes the classical weak Lebesgue space. Thus, it is also interesting to know whether or not this coincidence (1.2) remains true in the variable setting under some restriction on the variable exponent function.

In this article, we give positive answers to the above two questions. Indeed, in Theorem 1.5 below, we prove that the real interpolation space between the variable Hardy space and the space $L^{\infty}({{{\mathbb{R}}}^{n}})$ is just the variable weak Hardy space introduced in [31], via first establishing a useful decomposition for any distribution of the variable weak Hardy space into “good” and “bad” parts (see Proposition 2.1 below). As an application, we conclude that, when $p_{-}\in(1,\infty)$, the variable weak Hardy space ${W\!H^{p(\cdot)}({{{\mathbb{R}}}^{n}})}$ coincides with the variable weak Lebesgue space $W\!L^{p(\cdot)}({{{\mathbb{R}}}^{n}})$.

To state the main result of this article, we first recall some basic notions about the theory of real interpolation (see [5]). Let $(X_{0},X_{1})$ be a compatible couple of quasi-normed spaces, namely, $X_{0}$ and $X_{1}$ are two quasi-normed linear spaces which are continuously embedded into some large topological vector space. Let

[TABLE]

For any $t\in(0,\infty)$, the Peetre $K$-functional $K(t,f;X_{0},X_{1})$ on $X_{0}+X_{1}$ is defined by setting, for any $f\in X_{0}+X_{1}$,

[TABLE]

Then, for any $\theta\in(0,1)$ and $q\in(0,\infty]$, the real interpolation space $(X_{0},X_{1})_{\theta,q}$ between $X_{0}$ and $X_{1}$ is defined as

[TABLE]

where, for any $f\in X_{0}+X_{1}$,

[TABLE]

We also recall some notation about variable Lebesgue spaces. For a detailed exposition of these concepts, we refer the reader to the monographs [8, 12]. Denote by ${\mathcal{P}}({{{\mathbb{R}}}^{n}})$ the collection of all variable exponent functions $p(\cdot):\ {{{\mathbb{R}}}^{n}}\to(0,\infty)$ satisfying

[TABLE]

where $p_{-}$ is as in (1.1). For a measurable function $f$ on ${{{\mathbb{R}}}^{n}}$ and $p(\cdot)\in{\mathcal{P}}({{{\mathbb{R}}}^{n}})$, the modular functional (or, simply, the modular) $\varrho_{p(\cdot)}$, associated with $p(\cdot)$, is defined by setting

[TABLE]

and the Luxemburg (also known as the Luxemburg-Nakano) quasi-norm is given by setting

[TABLE]

Definition 1.1**.**

Let $p(\cdot)\in{\mathcal{P}}({{{\mathbb{R}}}^{n}})$.

1. (i)

The variable Lebesgue space ${L^{p(\cdot)}({{{\mathbb{R}}}^{n}})}$ is defined to be the set of all measurable functions $f$ on ${{{\mathbb{R}}}^{n}}$ such that the (quasi-)norm $\|f\|_{{L^{p(\cdot)}({{{\mathbb{R}}}^{n}})}}$ is finite. 2. (ii)

The variable weak Lebesgue space $W\!L^{p(\cdot)}({{{\mathbb{R}}}^{n}})$ is defined to be the set of all measurable functions $f$ on ${{{\mathbb{R}}}^{n}}$ such that

[TABLE]

Remark 1.2**.**

Let $p(\cdot)\in{\mathcal{P}}({{{\mathbb{R}}}^{n}})$ and $s\in(0,\infty)$.

1. (i)

It is easy to see that, for any $f\in{L^{p(\cdot)}({{{\mathbb{R}}}^{n}})}$, $\left\||f|^{s}\right\|_{{L^{p(\cdot)}({{{\mathbb{R}}}^{n}})}}=\|f\|_{L^{sp(\cdot)}({{{\mathbb{R}}}^{n}})}^{s}$. Moreover, for any $\lambda\in{\mathbb{C}}$ and $f,\ g\in{L^{p(\cdot)}({{{\mathbb{R}}}^{n}})}$, $\|\lambda f\|_{{L^{p(\cdot)}({{{\mathbb{R}}}^{n}})}}=|\lambda|\|f\|_{{L^{p(\cdot)}({{{\mathbb{R}}}^{n}})}}$ and

[TABLE]

here and hereafter,

[TABLE]

with $p_{-}$ as in (1.1). Particularly, when $p_{-}\in[1,\infty)$, ${L^{p(\cdot)}({{{\mathbb{R}}}^{n}})}$ is a Banach space (see [12, Theorem 3.2.7]). 2. (ii)

For any $f\in W\!L^{p(\cdot)}({{{\mathbb{R}}}^{n}})$, we have $\||f|^{s}\|_{W\!L^{p(\cdot)}({{{\mathbb{R}}}^{n}})}=\|f\|_{W\!L^{sp(\cdot)}({{{\mathbb{R}}}^{n}})}^{s}$ (see [31, Lemma 2.11]) and it was proved in [31, Lemma 2.9] that, for any $\lambda\in(0,\infty)$ and $f,\ g\in W\!L^{p(\cdot)}({{{\mathbb{R}}}^{n}})$, $\|\lambda f\|_{W\!L^{p(\cdot)}({{{\mathbb{R}}}^{n}})}=|\lambda|\|f\|_{W\!L^{p(\cdot)}({{{\mathbb{R}}}^{n}})}$ and

[TABLE]

A function $p(\cdot)\in{\mathcal{P}}({{{\mathbb{R}}}^{n}})$ is said to satisfy the globally log-Hölder continuous condition, denoted by $p(\cdot)\in C^{\log}({{{\mathbb{R}}}^{n}})$, if there exist positive constants $C_{\log}(p)$ and $C_{\infty}$, and $p_{\infty}\in{\mathbb{R}}$ such that, for any $x,\ y\in{{{\mathbb{R}}}^{n}}$,

[TABLE]

and

[TABLE]

In what follows, denote by ${\mathcal{S}}({{{\mathbb{R}}}^{n}})$ the space of all Schwartz functions on ${{{\mathbb{R}}}^{n}}$ equipped with the well-known classical topology and ${\mathcal{S}}^{\prime}({{{\mathbb{R}}}^{n}})$ its topological dual space equipped with the weak-$*$ topology. For any $N\in{\mathbb{N}}$, let

[TABLE]

where, for any $\beta:=(\beta_{1},\dots,\beta_{n})\in{\mathbb{Z}}_{+}^{n}$, $|\beta|:=\beta_{1}+\cdots+\beta_{n}$ and $D^{\beta}:=(\frac{\partial}{\partial x_{1}})^{\beta_{1}}\cdots(\frac{\partial}{\partial x_{n}})^{\beta_{n}}$. Then, for any $f\in{\mathcal{S}}^{\prime}({{{\mathbb{R}}}^{n}})$, the radial grand maximal function $f^{\ast}_{N,+}$ of $f$ is defined by setting, for any $x\in{{{\mathbb{R}}}^{n}}$,

[TABLE]

where, for any $t\in(0,\infty)$ and $\xi\in{{{\mathbb{R}}}^{n}}$, $\psi_{t}(\xi):=t^{-n}\psi(\xi/t)$.

Remark 1.3**.**

For any $a\in(0,\infty)$, $N\in{\mathbb{N}}$, $f\in{\mathcal{S}}^{\prime}({{{\mathbb{R}}}^{n}})$ and $x\in{{{\mathbb{R}}}^{n}}$, let

[TABLE]

Then, by an argument similar to that used in the proof of [36, Proposition 2.1], we know that $f_{N,+}^{\ast}\sim f_{N,\triangledown,a}^{\ast}$ with the equivalent positive constants independent of $f$.

Now we recall the definitions of both the variable Hardy space from Nakai and Sawano [24] and the variable weak Hardy space from Yan et al. [31].

Definition 1.4**.**

Let $p(\cdot)\in C^{\log}({{{\mathbb{R}}}^{n}})$ and $N\in(\frac{n}{\underline{p}}+n+1,\infty)$ be a positive integer, where $\underline{p}$ is as in (1.3).

1. (i)

The variable Hardy space ${H^{p(\cdot)}({{{\mathbb{R}}}^{n}})}$ is defined to be the set of all $f\in{\mathcal{S}}^{\prime}({{{\mathbb{R}}}^{n}})$ such that $f_{N,+}^{\ast}\in{L^{p(\cdot)}({{{\mathbb{R}}}^{n}})}$, equipped with the (quasi-)norm

[TABLE] 2. (ii)

The variable weak Hardy space ${W\!H^{p(\cdot)}({{{\mathbb{R}}}^{n}})}$ is defined to be the set of all $f\in{\mathcal{S}}^{\prime}({{{\mathbb{R}}}^{n}})$ such that $f_{N,+}^{\ast}\in W\!L^{p(\cdot)}({{{\mathbb{R}}}^{n}})$, equipped with the quasi-norm

[TABLE]

The main result of this article is stated as follows.

Theorem 1.5**.**

Let $p(\cdot)\in C^{\log}({{{\mathbb{R}}}^{n}})$ and $\theta\in(0,1)$. Then it holds true that

[TABLE]

where $\frac{1}{\widetilde{p}(\cdot)}=\frac{1-\theta}{p(\cdot)}$.

As a consequence of Theorem 1.5 and [24, Lemma 3.1], we immediately obtain the following conclusion.

Corollary 1.6**.**

Let $p(\cdot)\in C^{\log}({{{\mathbb{R}}}^{n}})$. If $p_{-}\in(1,\infty)$, then ${W\!H^{p(\cdot)}({{{\mathbb{R}}}^{n}})}=W\!L^{p(\cdot)}({{{\mathbb{R}}}^{n}})$ with equivalent quasi-norm.

Remark 1.7**.**

1. (i)

When $p(\cdot)\equiv p\in(0,1)$, Theorem 1.5 goes back to [15, Theorem 1], which states that

[TABLE] 2. (ii)

When $p(\cdot)\equiv 1$, (1.5) becomes

[TABLE]

which was presented in [26, (2)]. 3. (iii)

When $p(\cdot)\equiv p\in(1,\infty)$, (1.5) is a special case of [26, Theorem 7], namely,

[TABLE]

The proofs of Theorem 1.5 and Corollary 1.6 are presented in Section 3.

The main and key step in the proof of Theorem 1.5 is to decompose any distribution $f$ from the variable weak Hardy space $W\!H^{\widetilde{p}(\cdot)}({{{\mathbb{R}}}^{n}})$ into “good” and “bad” parts (see Proposition 2.1 below). The vector-valued inequality of the Hardy-Littlewood maximal function on the variable Lebesgue space, obtained by Cruz-Uribe and Fiorenza [9, Corollary 2.1] (see also Lemma 2.3 below), and the atomic characterization of $H^{p(\cdot)}({{{\mathbb{R}}}^{n}})$ via $(p(\cdot),\infty)$-atoms established by Nakai and Sawano [24, Theorem 4.5] (see also Lemma 2.5 below), play the key roles in the proof of Proposition 2.1. By using this proposition and some ideas from the proof of [21, Theorem 4.1], we further prove that

[TABLE]

where $\theta,\ p(\cdot)$ and $\widetilde{p}(\cdot)$ are as in Theorem 1.5. The converse part of (1.6) is proved by applying the sublinear operator $T(f):=f_{N,+}^{\ast}$, where $N$ is as in Definition 1.4, to the real interpolation property between $L^{p(\cdot)}({{{\mathbb{R}}}^{n}})$ and $L^{\infty}({{{\mathbb{R}}}^{n}})$, which is a special case of [21, Theorem 4.1] of Kempka and Vybíral when $p(\cdot)=q_{0}(\cdot)$ and $q=\infty$ (see also Lemma 3.1 below). Applying this last real interpolation property, between $L^{p(\cdot)}({{{\mathbb{R}}}^{n}})$ and $L^{\infty}({{{\mathbb{R}}}^{n}})$, and Theorem 1.5, we immediately obtain Corollary 1.6.

Here we point out that the approach used in the proof of (1.6) is quite different from that used in the proof of [15, Theorem 1] (see Remark 1.7(i) below). Indeed, in [15, Theorem 1], it seems to be only proved that the embedding

[TABLE]

holds true instead of $W\!H^{\widetilde{p}}({{{\mathbb{R}}}^{n}})\subset(H^{p}({{{\mathbb{R}}}^{n}}),L^{\infty}({{{\mathbb{R}}}^{n}}))_{\theta,\infty}$, since the proof of [15, Theorem 1] strongly depends on the decomposition obtained in [15, Lemma A] (see also Remark 2.2 below), which was proved only for any $f\in{\mathcal{S}}({{{\mathbb{R}}}^{n}})$ instead of all distributions and, more importantly, the Schwartz class is not dense in the space $W\!H^{\widetilde{p}}({{{\mathbb{R}}}^{n}})$ (see, for example, [18, 19]). To overcome this gap and difficulty, in this article, we establish a decomposition of any distribution of $f\in W\!H^{\widetilde{p}(\cdot)}({{{\mathbb{R}}}^{n}})$ into “good” and “bad” parts in Proposition 2.1 via a modification of technique due to Calderón [6] and some ideas from the proof of [31, Theorem 4.4] in which the atomic characterizations of variable weak Hardy spaces were obtained.

Finally, we make some conventions on notation. Let ${\mathbb{N}}:=\{1,2,\dots\}$, ${\mathbb{Z}}_{+}:={\mathbb{N}}\cup\{0\}$ and ${\mathbb{R}}_{+}^{n+1}:={{{\mathbb{R}}}^{n}}\times(0,\infty)$. We denote by $C$ a positive constant which is independent of the main parameters, but may vary from line to line. The symbol $f\lesssim g$ means $f\leq Cg$. If $f\lesssim g$ and $g\lesssim f$, we then write $f\sim g$. If $E$ is a subset of ${{{\mathbb{R}}}^{n}}$, we denote by $\chi_{E}$ its characteristic function and by $E^{\complement}$ the set ${{{\mathbb{R}}}^{n}}\backslash E$. For any $x\in{{{\mathbb{R}}}^{n}}$ and $r\in(0,\infty)$, denote by $Q(x,r)$ the cube centered at $x$ with side length $r$, whose sides are parallel to the axes of coordinates. For each cube $Q\subset{{{\mathbb{R}}}^{n}}$, we use $x_{Q}$ to denote its center and $\ell(Q)$ its side length and, for any $a\in(0,\infty)$, denote by $aQ$ the cube concentric with $Q$ having the side length $a\ell(Q)$. We use $\vec{0}_{n}$ to denote the origin of ${{{\mathbb{R}}}^{n}}$. For any $a\in{\mathbb{R}}$, let $\lfloor a\rfloor$ be the maximal integer not bigger than $a$.

2 A key decomposition

In this section, we aim to establish a decomposition of any distribution $f$ belonging to $W\!H^{p(\cdot)/(1-\theta)}({{{\mathbb{R}}}^{n}})$ into “good” and “bad” parts in Proposition 2.1 below, which plays a key role in the proofs of Theorem 1.5 and Corollary 1.6 in Section 3.

We begin with some notation. For any $p(\cdot)\in{\mathcal{P}}({{{\mathbb{R}}}^{n}})$, let $\psi\in{\mathcal{S}}({{{\mathbb{R}}}^{n}})$ be such that ${\mathop{\mathrm{missing}}{\,supp\,}}\psi\subset B(\vec{0}_{n},1)$, $\int_{{{\mathbb{R}}}^{n}}\psi(x)x^{\gamma}\,dx=0$ for any $\gamma\in{\mathbb{Z}}_{+}^{n}$ with $|\gamma|\leq s_{0}$, where

[TABLE]

Then there exists $\phi\in{\mathcal{S}}({{{\mathbb{R}}}^{n}})$ satisfying that $\widehat{\phi}$ has compact support away from the origin and, for any $x\in{{{\mathbb{R}}}^{n}}\backslash\{\vec{0}_{n}\}$,

[TABLE]

see, for example, [6, (3.1)]. Define a function $\eta$ on ${{{\mathbb{R}}}^{n}}$ by setting, for any $x\in{{{\mathbb{R}}}^{n}}\backslash\{\vec{0}_{n}\}$,

[TABLE]

and $\widehat{\eta}(\vec{0}_{n}):=1$. Then $\eta$ is infinitely differentiable, has compact support and equals 1 near the origin (see [6, p. 219]). Moreover, for any $t_{0},\ t_{1}\in(0,\infty)$ and $x\in{{{\mathbb{R}}}^{n}}\backslash\{\vec{0}_{n}\}$,

[TABLE]

Let $x_{0}:=(\overbrace{2,\,...\,,2}^{n\,{\rm times}})\in{{{\mathbb{R}}}^{n}}$ and $f\in{W\!H^{p(\cdot)}({{{\mathbb{R}}}^{n}})}$. For any $x\in{{{\mathbb{R}}}^{n}}$, let

[TABLE]

and, for any $t\in(0,\infty)$,

[TABLE]

Then, using (2.2), we have, for any $t_{0},\ t_{1}\in(0,\infty)$ and $x\in{{{\mathbb{R}}}^{n}}\backslash\{\vec{0}_{n}\}$,

[TABLE]

and, by [6, p. 220] and the fact that $f\in{\mathcal{S}}^{\prime}({{{\mathbb{R}}}^{n}})$, we know that

[TABLE]

converges in ${\mathcal{S}}^{\prime}({{{\mathbb{R}}}^{n}})$. Now, for any $x\in{{{\mathbb{R}}}^{n}}$, let

[TABLE]

Then $M_{\triangledown}(f)$ is lower semi-continuous and, by Remark 1.3 and [31, Corollary 3.8], we further know that $M_{\triangledown}(f)\in W\!L^{p(\cdot)}({{{\mathbb{R}}}^{n}})$ and, moreover,

[TABLE]

with the implicit equivalent positive constants independent of $f$.

Now we have the following decomposition for elements of the variable weak Hardy space.

Proposition 2.1**.**

Assume that $\theta\in(0,1)$, $p(\cdot)$ and $\widetilde{p}(\cdot)$ are as in Theorem 1.5. Let $f\in W\!H^{\widetilde{p}(\cdot)}({{{\mathbb{R}}}^{n}})$ and $\alpha\in(0,\infty)$. Then there exist $g_{\alpha}\in L^{\infty}({{{\mathbb{R}}}^{n}})$ and $b_{\alpha}\in{\mathcal{S}}^{\prime}({{{\mathbb{R}}}^{n}})$ such that $f=g_{\alpha}+b_{\alpha}$ in ${\mathcal{S}}^{\prime}({{{\mathbb{R}}}^{n}})$, $\|g_{\alpha}\|_{L^{\infty}({{{\mathbb{R}}}^{n}})}\leq C_{1}\alpha$ and

[TABLE]

where $C_{1}$ and $C_{2}$ are two positive constants independent of $f$ and $\alpha$.

In what follows, to simplify the presentation of this article, for any measurable function $g$ and $\alpha,\,\beta\in(0,\infty)$ with $\alpha<\beta$, we always write $\chi_{\{x\in{{{\mathbb{R}}}^{n}}:\ g(x)>\alpha\}}$ or $\chi_{\{x\in{{{\mathbb{R}}}^{n}}:\ \alpha<g(x)\leq\beta\}}$ simply by $\chi_{\{g>\alpha\}}$ or $\chi_{\{\alpha<g\leq\beta\}}$.

Remark 2.2**.**

It was established in [15, Lemma A] that, if $f\in{\mathcal{S}}({{{\mathbb{R}}}^{n}})$ and $\alpha\in(0,\infty)$, then $f$ can be written as the sum of two functions, $g$ and $b$, such that $\|g\|_{L^{\infty}({{{\mathbb{R}}}^{n}})}\leq\widetilde{C}_{1}\alpha$ and

[TABLE]

where $C_{1}$ and $C_{2}$ are two positive constants independent of $\alpha$ and $f$. Comparing with [15, Lemma A], Proposition 2.1 presents a decomposition of any distribution from $W\!H^{\widetilde{p}(\cdot)}({{{\mathbb{R}}}^{n}})$ and, in this sense, it is a very useful improvement of [15, Lemma A].

To prove Proposition 2.1, we need the following vector-valued inequality of the Hardy-Littlewood maximal operator ${\mathcal{M}}$ on the variable Lebesgue space, which was obtained by Cruz-Uribe and Fiorenza [9, Corollary 2.1]. Here and hereafter, the operator ${\mathcal{M}}$ is defined by setting, for any locally integrable function $f$ on ${{{\mathbb{R}}}^{n}}$ and $x\in{{{\mathbb{R}}}^{n}}$,

[TABLE]

where the supremum is taken over all balls $B$ of ${{{\mathbb{R}}}^{n}}$ containing $x$.

Lemma 2.3**.**

Let $r\in(1,\infty)$. Assume that $p(\cdot)\in C^{\log}({{{\mathbb{R}}}^{n}})$ satisfies $1<p_{-}\leq p_{+}<\infty$. Then there exists a positive constant $C$ such that, for any sequence $\{f_{j}\}_{j\in{\mathbb{N}}}$ of measurable functions,

[TABLE]

The atomic characterization of ${H^{p(\cdot)}({{{\mathbb{R}}}^{n}})}$, obtained by Nakai and Sawano [24], also plays a key role in the proof of Proposition 2.1. The following notions of $(p(\cdot),\infty)$-atoms and the atomic Hardy space $H_{\rm atom}^{p(\cdot),\infty}({{{\mathbb{R}}}^{n}})$ come from [24, Definition 1.4] and [24, Definition 1.5] of Nakai and Sawano, respectively.

Definition 2.4**.**

Let $p(\cdot)\in C^{\log}({{{\mathbb{R}}}^{n}})$.

1. (i)

A measurable function $a$ on ${{{\mathbb{R}}}^{n}}$ is called a $(p(\cdot),\infty)$-atom if there exists a cube $Q\subset{{{\mathbb{R}}}^{n}}$ such that ${\mathop{\mathrm{missing}}{\,supp\,}}a\subset Q$,

[TABLE]

and $\int_{{{\mathbb{R}}}^{n}}a(x)x^{\alpha}\,dx=0$ for any $\alpha\in{\mathbb{Z}}_{+}^{n}$ with $|\alpha|\leq\lfloor n(\frac{1}{p_{-}}-1)\rfloor$. 2. (ii)

The variable atomic Hardy space $H_{\rm atom}^{p(\cdot),\infty}({{{\mathbb{R}}}^{n}})$ is defined as the space of all $f\in{\mathcal{S}}^{\prime}({{{\mathbb{R}}}^{n}})$ such that $f=\sum_{j\in{\mathbb{N}}}\lambda_{j}a_{j}$ in ${\mathcal{S}}^{\prime}({{{\mathbb{R}}}^{n}})$, where $\{\lambda_{j}\}_{j\in{\mathbb{N}}}$ is a sequence of non-negative numbers and $\{a_{j}\}_{j\in{\mathbb{N}}}$ is a sequence of $(p(\cdot),\infty)$-atoms, associated with cubes $\{Q_{j}\}_{j\in{\mathbb{N}}}$ of ${{{\mathbb{R}}}^{n}}$. Moreover, for any $f\in H_{\rm atom}^{p(\cdot),\infty}({{{\mathbb{R}}}^{n}})$, let

[TABLE]

where $\underline{p}$ is as in (1.3) and the infimum is taken over all admissible decompositions of $f$ as above.

The following atomic characterization of ${H^{p(\cdot)}({{{\mathbb{R}}}^{n}})}$ by means of $(p(\cdot),\infty)$-atoms is a part of [24, Theorem 4.5] of Nakai and Sawano.

Lemma 2.5**.**

Let $p(\cdot)\in C^{\log}({{{\mathbb{R}}}^{n}})$. Then ${H^{p(\cdot)}({{{\mathbb{R}}}^{n}})}=H_{\rm atom}^{p(\cdot),\infty}({{{\mathbb{R}}}^{n}})$ with equivalent quasi-norms.

Now we can show Proposition 2.1 by using Lemmas 2.3 and 2.5 as follows.

Proof of Proposition 2.1.

Let $f\in W\!H^{\widetilde{p}(\cdot)}({{{\mathbb{R}}}^{n}})$ and, for any $i\in{\mathbb{Z}}$,

[TABLE]

Then ${\Omega}_{i}$ is open and, by (2.6), we find that

[TABLE]

By the Whitney decomposition (see, for example, [17, p. 463]), we know that, for any $i\in{\mathbb{Z}}$, there exists a sequence $\{{Q_{i,j}}\}_{j\in{\mathbb{N}}}$ of cubes such that

1. (i)

$\bigcup_{j\in{\mathbb{N}}}{Q_{i,j}}={\Omega}_{i}$ and $\{{Q_{i,j}}\}_{j\in{\mathbb{N}}}$ have disjoint interiors; 2. (ii)

for any $j\in{\mathbb{N}}$, $\sqrt{n}\,l_{{Q_{i,j}}}\leq{\mathop{\mathrm{missing}}{\,dist\,}}({Q_{i,j}},\,{\Omega}_{i}^{\complement})\leq 4\sqrt{n}\,l_{{Q_{i,j}}}$, where $l_{{Q_{i,j}}}$ denotes the side length of the cube ${Q_{i,j}}$ and ${\mathop{\mathrm{missing}}{\,dist\,}}({Q_{i,j}},\,{\Omega}_{i}^{\complement}):=\inf\{|x-y|:\ x\in{Q_{i,j}},\ y\in{\Omega}_{i}^{\complement}\}$; 3. (iii)

for any $j,\,k\in{\mathbb{N}}$, if the boundaries of cubes ${Q_{i,j}}$ and $Q_{i,k}$ touch, then $\frac{1}{4}\leq\frac{l_{{Q_{i,j}}}}{l_{Q_{i,k}}}\leq 4;$ 4. (iv)

for any given $j\in{\mathbb{N}}$, there exist at most $12n$ different cubes $Q_{i,k}$ that touch ${Q_{i,j}}$.

For any $i\in{\mathbb{Z}}$, $j\in{\mathbb{N}}$ and $x\in{{{\mathbb{R}}}^{n}}$, let

[TABLE]

[TABLE]

[TABLE]

and, for any $\epsilon\in(0,\infty)$,

[TABLE]

and

[TABLE]

where

[TABLE]

It is easy to see that, for any $x\in{{{\mathbb{R}}}^{n}}$, $\{{\mathop{\mathrm{missing}}{\,dist\,}}(x,\,{\Omega}_{i}^{\complement})\}_{i\in{\mathbb{Z}}}$ is decreasing in $i$, since $\{\Omega_{i}\}_{i\in{\mathbb{Z}}}$ is decreasing in $i$.

Next we finish the proof of this proposition by fours steps.

Step 1) In this step, we show that, for any $\epsilon\in(0,\infty)$ and $i\in{\mathbb{Z}}$,

[TABLE]

with the implicit positive constant independent of $\epsilon$, $i$ and $f$.

To this end, we first observe that, for any given $(x,t)\in{\mathbb{R}}_{+}^{n+1}$,

[TABLE]

For any $i\in{\mathbb{Z}}$, let $\Omega_{i}^{\ast,1}:=\{(y,t)\in{\mathbb{R}}_{+}^{n+1}:\ y\in\Omega_{i},\ (y,t)\in\widetilde{\Omega}_{i}\}$ and

[TABLE]

Then $\chi_{\Omega_{i}^{\ast}}=\chi_{\Omega_{i}^{\ast,1}}\chi_{\Omega_{i}^{\ast,2}}$.

If $t\geq\frac{{\mathop{\mathrm{missing}}{\,dist\,}}(x,\Omega_{i}^{\complement})}{|x_{0}|+1}$, then, for any $(y,t)\in{\mathbb{R}}_{+}^{n+1}$ with $y\in B(x,t[|x_{0}|+1])$,

[TABLE]

which implies that $(y,t)\notin\widetilde{\Omega}_{i}$ and hence $\chi_{\Omega_{i}^{\ast,1}}(y,t)\widetilde{\psi}_{t}(x-y)=0$.

If $t<\frac{{\mathop{\mathrm{missing}}{\,dist\,}}(x,\Omega_{i}^{\complement})}{3(|x_{0}|+1)}$, then, for any $(y,t)\in{\mathbb{R}}_{+}^{n+1}$ with $y\in B(x,t[|x_{0}|+1])$,

[TABLE]

which implies that $(y,t)\in\widetilde{\Omega}_{i}$, $y\in\Omega_{i}$ and hence $\chi_{\Omega_{i}^{\ast,1}}(y,t)\widetilde{\psi}_{t}(x-y)=\widetilde{\psi}_{t}(x-y)$.

If $t\geq\frac{{\mathop{\mathrm{missing}}{\,dist\,}}(x,\Omega_{i+1}^{\complement})}{|x_{0}|+1}$, then, for any $(y,t)\in{\mathbb{R}}_{+}^{n+1}$ with $y\in B(x,t[|x_{0}|+1])\cap\Omega_{i}$,

[TABLE]

which implies that $(y,t)\notin\widetilde{\Omega}_{i+1}$ and hence $\chi_{\Omega_{i+1}^{\ast,2}}(y,t)\widetilde{\psi}_{t}(x-y)=\widetilde{\psi}_{t}(x-y)$.

If $t<\frac{{\mathop{\mathrm{missing}}{\,dist\,}}(x,\Omega_{i+1}^{\complement})}{3(|x_{0}|+1)}$, then, for any $(y,t)\in{\mathbb{R}}_{+}^{n+1}$ with $y\in B(x,t[|x_{0}|+1])$,

[TABLE]

which implies that $(y,t)\in\widetilde{\Omega}_{i+1}$ and hence $\chi_{\Omega_{i+1}^{\ast,1}}(y,t)\widetilde{\psi}_{t}(x-y)=0$.

From the above arguments, we deduce that, for any given $(x,t)\in{\mathbb{R}}_{+}^{n+1}$, if

[TABLE]

then, for any $y\in{{{\mathbb{R}}}^{n}}$, it holds true that

[TABLE]

Next, for any given $i\in{\mathbb{Z}}$ and $x\in{{{\mathbb{R}}}^{n}}$, we estimate $|b_{i}^{\epsilon}(x)|$ in two cases.

Case 1.1) $\frac{{\mathop{\mathrm{missing}}{\,dist\,}}(x,\Omega_{i}^{\complement})}{3(|x_{0}|+1)}\leq\frac{{\mathop{\mathrm{missing}}{\,dist\,}}(x,\Omega_{i+1}^{\complement})}{|x_{0}|+1}$.

In this case, we have

[TABLE]

Then, by (2.11), we know that

[TABLE]

Since $\chi_{\Omega_{i}^{\ast}}(y,t)\neq 0$ implies that $(y,t)\notin\widetilde{\Omega}_{i+1}$, it follows that $|F(y,t)|\leq 2^{i+1}$, where $F$ is as in (2.3). By this, (2.12) and the fact that, for any $t\in(0,\infty)$ and $x\in{{{\mathbb{R}}}^{n}}$, $\int_{{{\mathbb{R}}}^{n}}|\widetilde{\psi}_{t}(x-y)|\,dy\lesssim 1$, we conclude that

[TABLE]

with the implicit positive constants independent of $\epsilon$, $i$ and $x$.

Case 1.2) $\frac{{\mathop{\mathrm{missing}}{\,dist\,}}(x,\Omega_{i}^{\complement})}{3(|x_{0}|+1)}>\frac{{\mathop{\mathrm{missing}}{\,dist\,}}(x,\Omega_{i+1}^{\complement})}{|x_{0}|+1}$.

In this case, we have

[TABLE]

Then, by (2.9), (2.10) and (2.11), we obtain

[TABLE]

By the argument same as that used in the proof of (2.13), we have

[TABLE]

For ${\rm I}_{3}$, by (2.4), we find that

[TABLE]

On the other hand, if $t\in[\frac{{\mathop{\mathrm{missing}}{\,dist\,}}(x,\Omega_{i+1}^{\complement})}{|x_{0}|+1},\frac{{\mathop{\mathrm{missing}}{\,dist\,}}(x,\Omega_{i}^{\complement})}{3(|x_{0}|+1)}]$, then ${\mathop{\mathrm{missing}}{\,dist\,}}(x,\Omega_{i+1}^{\complement})\leq t(|x_{0}|+1)$. Thus, there exists $x^{\ast}\in\Omega_{i+1}^{\complement}$ such that $|x-x^{\ast}|\leq 3t(|x_{0}|+1)$, which, together with (2.5), further implies that

[TABLE]

By this and (2.14), we find that ${\rm I}_{3}\leq 2^{i}$. Therefore, in this case, we also have $|b_{i}^{\epsilon}(x)|\lesssim 2^{i}$, which, combined with (2.13), implies that (2.8) holds true. This finishes the proof of Step 1).

Step 2) In this step, we construct $g_{\alpha}$ for any $\alpha\in(0,\infty)$.

By (2.8) of Step 1), we conclude that, for any $i\in{\mathbb{Z}}$, $\{b_{i}^{\epsilon}\}_{\epsilon\in(0,\infty)}$ is bounded in $L^{\infty}({{{\mathbb{R}}}^{n}})$ uniformly with respect to $\epsilon\in(0,\infty)$. Thus, by the Alaoglu theorem (see, for example, [27, Theorem 3.17]) and the well-known diagonal rule, we find that there exist $\{b_{i}\}_{i\in{\mathbb{Z}}}\subset L^{\infty}({{{\mathbb{R}}}^{n}})$ and $\{\epsilon_{k}\}_{k\in{\mathbb{N}}}\subset(0,\infty)$ such that $\epsilon_{k}\to 0$ as $k\to\infty$ and, for any $i\in{\mathbb{Z}}$ and $g\in L^{1}({{{\mathbb{R}}}^{n}})$,

[TABLE]

moreover, for any $i\in{\mathbb{Z}}$,

[TABLE]

Next, we claim that, for any $i_{0}\in{\mathbb{Z}}$,

[TABLE]

Indeed, by (2.8), we know that, for any $\epsilon\in(0,\infty)$,

[TABLE]

and, by (2.16), we have

[TABLE]

Thus, both $\sum_{i=-\infty}^{i_{0}}b_{i}^{\epsilon}$ and $\sum_{i=-\infty}^{i_{0}}b_{i}$ converge in $L^{\infty}({{{\mathbb{R}}}^{n}})$ and, for any $g\in{\mathcal{S}}({{{\mathbb{R}}}^{n}})$,

[TABLE]

Therefore, for any $\delta\in(0,\infty)$, there exists $N\in{\mathbb{N}}\cap[|i_{0}|,\infty)$, depending on $\delta$ and $n$, such that

[TABLE]

On the other hand, by (2.15), we find that, for any given $i\in{\mathbb{Z}}$ and $g\in{\mathcal{S}}({{{\mathbb{R}}}^{n}})$, there exists $K_{i}\in{\mathbb{N}}$, depending on $i$ and $g$, such that, when $k>K_{i}$,

[TABLE]

Let $K:=\max\{K_{-N+1},K_{-N+2},\dots,K_{i_{0}}\}$. Then, for any given $i\in{\mathbb{Z}}$ and $g\in{\mathcal{S}}({{{\mathbb{R}}}^{n}})$, by (2.18) and (2.19), we know that, for any $k>K$,

[TABLE]

where the implicit positive constant depends on $i$ and $g$. This implies that (2.17) holds true.

Now, for any given $\alpha\in(0,\infty)$, we choose $i_{0}\in{\mathbb{Z}}$ such that $2^{i_{0}}\leq\alpha<2^{i_{0}+1}$. Let

[TABLE]

Then, by the above claim and (2.19), we conclude that $g_{\alpha}\in L^{\infty}({{{\mathbb{R}}}^{n}})$ and

[TABLE]

which completes the proof of Step 2).

Step 3) In this step, we construct $b_{\alpha}$ for any $\alpha\in(0,\infty)$.

From an argument similar to that used in the proof of [31, Theorem 4.4], we deduce that there exist $\{b_{i,j}\}_{i>i_{0},j\in{\mathbb{N}}}\subset L^{\infty}({{{\mathbb{R}}}^{n}})$ and a subsequence $\{\epsilon_{k_{l}}\}_{l\in{\mathbb{N}}}\subset\{\epsilon_{k}\}_{k\in{\mathbb{N}}}$ such that $\epsilon_{k_{l}}\to 0$ as $l\to\infty$ and, for any $i>i_{0}$, $j\in{\mathbb{N}}$ and $g\in L^{1}({{{\mathbb{R}}}^{n}})$,

[TABLE]

${\mathop{\mathrm{missing}}{\,supp\,}}b_{i,j}\subset c_{1}Q_{i,j}$ for some constant $c_{1}\in(1,\infty)$, $\|b_{i,j}\|_{L^{\infty}({{{\mathbb{R}}}^{n}})}\lesssim 2^{i}$ and $\int_{{{\mathbb{R}}}^{n}}b_{i,j}(x)x^{\gamma}\,dx=0$ for any $\gamma\in{\mathbb{Z}}_{+}^{n}$ with $|\gamma|\leq s_{0}$, where $s_{0}$ is as in (2.1). Moreover,

[TABLE]

Let

[TABLE]

Next, we show that $b_{\alpha}$ satisfies (2.7). By the construction of $b_{i,j}$, we know that there exists $\widetilde{c}_{1}\in(0,\infty)$ such that

[TABLE]

is a linear combination of $(p(\cdot),\infty)$-atoms (see Definition 2.4(i)). Then, by Lemma 2.3, the fact that $\chi_{c_{1}Q_{i,j}}\lesssim{\mathcal{M}}(\chi_{Q_{i,j}})$ and property (i) of the previous Whitney decomposition presented in this proof, we find that

[TABLE]

where $\underline{p}$ is as in (1.3). On the other hand, by the definition of $W\!L^{\widetilde{p}(\cdot)}({{{\mathbb{R}}}^{n}})$ and (2.6), we know that, for $\theta\in(0,1)$,

[TABLE]

Combining (2.20), (2.21) and (2.22), we conclude that $b_{\alpha}$ belongs to the variable atomic Hardy space $H_{\rm atom}^{p(\cdot),\infty}({{{\mathbb{R}}}^{n}})$ (see Definition 2.4(ii)). By this and Lemma 2.5, we find that

[TABLE]

This finishes the proof of Step 3).

Step 4) In this step, we prove that $f=g_{\alpha}+b_{\alpha}$ in ${\mathcal{S}}^{\prime}({{{\mathbb{R}}}^{n}})$.

We first claim that, for any given $i\in{\mathbb{Z}}$, $\epsilon\in(0,\infty)$ and $x\in{{{\mathbb{R}}}^{n}}$,

[TABLE]

Indeed, since, for any $(y,t)\in{{{\mathbb{R}}}^{n}}\times(0,\infty)$ and all $x\in B(y,3[|x_{0}|+1]t)$, $F(y,t)\leq M_{\triangledown}(f)(x)$, it follows that

[TABLE]

which, together with the fact that, for any measurable subset $E\subset{{{\mathbb{R}}}^{n}}$,

[TABLE]

implies that

[TABLE]

Thus, by (2.24), $\sum_{j\in{\mathbb{N}}}\chi_{\widetilde{Q}_{i,j}}=\chi_{\Omega_{i}^{\ast}}\leq 1$ and the Lebesgue dominated convergence theorem, we conclude that

[TABLE]

Therefore, (2.23) holds true.

Now, by the argument same as that used in [31, p. 2855], (2.17) and (2.23), we find that

[TABLE]

where all above summations converge in ${\mathcal{S}}^{\prime}({{{\mathbb{R}}}^{n}})$. This finishes the proof of Step 4) and hence the proof of Proposition 2.1. ∎

3 Proofs of Theorem 1.5 and Corollary 1.6

In this section, we prove Theorem 1.5 and Corollary 1.6 by using Proposition 2.1. To this end, we need the following known real interpolation result, which is [21, Theorem 4.1] of Kempka and Vybíral in the case when $p(\cdot)=q_{0}(\cdot)$ and $q=\infty$.

Lemma 3.1**.**

Let $p(\cdot)\in{\mathcal{P}}({{{\mathbb{R}}}^{n}})$ and $\theta\in(0,1)$. Then it holds true that

[TABLE]

Using Proposition 2.1 and Lemma 3.1, we now show Theorem 1.5 as follows.

Proof of Theorem 1.5.

We first prove that

[TABLE]

Let $f\in W\!H^{\widetilde{p}(\cdot)}({{{\mathbb{R}}}^{n}})$. Then we show $f\in(H^{p(\cdot)}({{{\mathbb{R}}}^{n}}),L^{\infty}({{{\mathbb{R}}}^{n}}))_{\theta,\infty}$ by two steps.

Step 1) In this step, we estimate $K(t,f;H^{p(\cdot)}({{{\mathbb{R}}}^{n}}),L^{\infty}({{{\mathbb{R}}}^{n}}))$ for any $t\in(0,\infty)$.

By Proposition 2.1, we know that, for any $\alpha\in(0,\infty)$, there exist $g_{\alpha}\in L^{\infty}({{{\mathbb{R}}}^{n}})$ and $b_{\alpha}\in H^{p(\cdot)}({{{\mathbb{R}}}^{n}})$ such that $f=g_{\alpha}+b_{\alpha}$ in ${\mathcal{S}}^{\prime}({{{\mathbb{R}}}^{n}})$, $\|g_{\alpha}\|_{L^{\infty}({{{\mathbb{R}}}^{n}})}\lesssim\alpha$ and $b_{\alpha}$ satisfies (2.7). Then, by Proposition 2.1, we find that, for any $t\in(0,\infty)$,

[TABLE]

which, together with Remark 1.2(i), implies that

[TABLE]

where, for any $\lambda\in(0,\infty)$,

[TABLE]

For any $t\in(0,\infty)$, let

[TABLE]

Observe that the function $h$ is decreasing on $(0,\infty)$. Then it is easy to see that

[TABLE]

From this and (3.2), we deduce that

[TABLE]

Step 2) In this step, we estimate

[TABLE]

By (3.3), we find that

[TABLE]

Notice that, when $2^{k}<\alpha(t)$,

[TABLE]

which, combined with (3.4), implies that

[TABLE]

If $\frac{1-\theta}{\underline{p}}\leq 1$ with $\underline{p}$ as in (1.3), then, by the well-known inequality that, for any $d\in(0,1]$ and $\{a_{i}\}_{i\in{\mathbb{N}}}\subset{\mathbb{C}}$,

[TABLE]

we know that

[TABLE]

If $\frac{1-\theta}{\underline{p}}>1$ with $\underline{p}$ as in (1.3), then, by the Hölder inequality, we find that, for any $\epsilon\in(0,\frac{\theta}{1-\theta})$,

[TABLE]

Now, by (3.5) and (3.6), we conclude that

[TABLE]

which completes the proof of Step 2). Therefore, $f\in(H^{p(\cdot)}({{{\mathbb{R}}}^{n}}),L^{\infty}({{{\mathbb{R}}}^{n}}))_{\theta,\infty}$ and hence (3.1) holds true.

Conversely, we need to show that

[TABLE]

To prove (3.7), let $T$ be a sublinear operator defined by setting, for any $f\in{\mathcal{S}}^{\prime}({{{\mathbb{R}}}^{n}})$, $T(f):=f_{N,+}^{\ast}$, where $N$ is as in Definition 1.4 and $f_{N,+}^{\ast}$ is as in (1.4).

We claim that the operator $T$ is bounded from the space $({H^{p(\cdot)}({{{\mathbb{R}}}^{n}})},L^{\infty}({{{\mathbb{R}}}^{n}}))_{\theta,\infty}$ to the space $({L^{p(\cdot)}({{{\mathbb{R}}}^{n}})},L^{\infty}({{{\mathbb{R}}}^{n}}))_{\theta,\infty}$. Indeed, let $g\in({H^{p(\cdot)}({{{\mathbb{R}}}^{n}})},L^{\infty}({{{\mathbb{R}}}^{n}}))_{\theta,\infty}$. Then, by the definition of $({H^{p(\cdot)}({{{\mathbb{R}}}^{n}})},L^{\infty}({{{\mathbb{R}}}^{n}}))_{\theta,\infty}$, we know that there exist $g_{0}\in{H^{p(\cdot)}({{{\mathbb{R}}}^{n}})}$ and $g_{1}\in L^{\infty}({{{\mathbb{R}}}^{n}})$ such that

[TABLE]

Moreover, observe that $T(g)\leq T(g_{0})+T(g_{1})$. Notice that $T$ is bounded from $L^{\infty}({{{\mathbb{R}}}^{n}})$ to $L^{\infty}({{{\mathbb{R}}}^{n}})$ and also from ${H^{p(\cdot)}({{{\mathbb{R}}}^{n}})}$ to ${L^{p(\cdot)}({{{\mathbb{R}}}^{n}})}$. It follows that $T(g_{0})\in{L^{p(\cdot)}({{{\mathbb{R}}}^{n}})}$ and $T(g_{1})\in L^{\infty}({{{\mathbb{R}}}^{n}})$. Let

[TABLE]

Then ${{{\mathbb{R}}}^{n}}=[E_{0}\cup E_{1}]=[E_{0}\cup(E_{1}\backslash E_{0})]$. Thus, we have

[TABLE]

From this and (3.8), we deduce that

[TABLE]

Therefore, the above claim holds true.

By this claim and Lemma 3.1, we conclude that, if $f\in({H^{p(\cdot)}({{{\mathbb{R}}}^{n}})},L^{\infty}({{{\mathbb{R}}}^{n}}))_{\theta,\infty}$, then $T(f)$ belongs to $W\!L^{\widetilde{p}(\cdot)}({{{\mathbb{R}}}^{n}})$, namely, $f\in W\!H^{\widetilde{p}(\cdot)}({{{\mathbb{R}}}^{n}})$. Thus, (3.7) holds true. This finishes the proof of Theorem 1.5. ∎

We end this section by giving the proof of Corollary 1.6 via using Theorem 1.5 and Lemma 3.1.

Proof of Corollary 1.6.

Since $p_{-}\in(1,\infty)$, it follows from [24, Lemma 3.1] that

[TABLE]

with equivalent norms. Moreover, there exists $\theta\in(0,1)$ such that $(1-\theta)p_{-}\in(1,\infty)$. By this, (3.9), Theorem 1.5 and the fact that

[TABLE]

(see Lemma 3.1), we conclude that

[TABLE]

This finishes the proof of Corollary 1.6. ∎