Local Muckenhoupt class for variable exponents

Abstract

This work extends the theory of Rychkov, who developed the theory of $A_{p}^{\mathrm{loc}}$ weights. It also extends the work by Cruz-Uribe SFO, Fiorenza, and Neugebauer. The class $A_{p(\cdot )}^{\mathrm{loc}}$ is defined. The weighted inequality for the local Hardy–Littlewood maximal operator on Lebesgue spaces with variable exponents is proven. Cruz-Uribe SFO, Fiorenza, and Neugebauer considered the Muckenhoupt class for Lebesgue spaces with variable exponents. However, due to the setting of variable exponents, a new method for extending weights is needed. The proposed extension method differs from that by Rychkov. A passage to the vector-valued inequality is realized by means of the extrapolation technique. This technique is an adaptation of the work by Cruz-Uribe and Wang. Additionally, a theory of extrapolation adapted to our class of weights is also obtained.

1 Introduction

This paper develops the theory of local Muckenhoupt weights in a variable exponent setting. It mixes the results obtained in [2, 3, 14]. Also see the textbook [6]. Due to the setting of variable exponents, we cannot directly use the ideas of Cruz-Uribe, Diening, and Hästö [2], Cruz-Uribe SFO, Fiorenza and Neugebauer [3], or Rychkov [14].

Herein, we use the following notation of variable exponents: Let $p(\cdot ):\mathbb{R}^{n}\to [1,\infty )$ be a measurable function, and let w be a weight. In other words, $w:\mathbb{R}^{n}\to [1,\infty )$ is a measurable function that is positive almost everywhere. Then the weighted variable Lebesgue space $L^{p(\cdot )}(w)$ collects all measurable functions f such that

$$ \int _{\mathbb{R}^{n}} \biggl(\frac{ \vert f(x) \vert }{\lambda } \biggr)^{p(x)}w(x) {\,\mathrm{d}}x< \infty $$

for some $\lambda >0$. For $f \in L^{p(\cdot )}(w)$, the norm is defined by

$$ \Vert f \Vert _{L^{p(\cdot )}(w)} \equiv \inf \biggl\{ \lambda >0: \int _{ \mathbb{R}^{n}} \biggl(\frac{ \vert f(x) \vert }{\lambda } \biggr)^{p(x)}w(x) { \,\mathrm{d}}x\le 1 \biggr\} . $$

If $w\equiv 1$, then $\|\cdot \|_{L^{p(\cdot )}(1)}=\|\cdot \|_{p(\cdot )}$ and $L^{p(\cdot )}(1)=L^{p(\cdot )}({\mathbb{R}}^{n})$. Thus, we have the ordinary variable Lebesgue space $L^{p(\cdot )}({\mathbb{R}}^{n})$.

The definition of $L^{p(\cdot )}(w)$ slightly differs from that in [3], where the authors considered the theory of Muckenhoupt weights for the Hardy–Littlewood maximal operator M for Lebesgue spaces with variable exponents. Recall that Rychkov established the theory of the local Muckenhoupt class [14]. Here and below, $\mathcal{Q}$ denotes the set of all cubes whose edges are parallel to the coordinate axes. Herein we mix the notions considered in [3, 14] to define the local Muckenhoupt class with variable exponents.

Definition 1.1

Given an exponent $p(\cdot ):\mathbb{R}^{n} \to [1,\infty )$, a weight w belongs to $A^{\mathrm{loc}}_{p(\cdot )}$ if $[w]_{A^{\mathrm{loc}}_{p(\cdot )}} \equiv \sup_{Q \in \mathcal{Q}, |Q|\le 1} |Q|^{-1} \|\chi _{Q}\|_{L^{p(\cdot )}(w)}\| \chi _{Q}\|_{L^{p'(\cdot )}(\sigma )}<\infty$, where $\sigma \equiv w^{-\frac{1}{p(\cdot )-1}}$ and the supremum is taken over all cubes $Q \in \mathcal{Q}$ with $|Q|\le 1$. Given a cube Q, analogously define $A^{\mathrm{loc}}_{p(\cdot )}(Q)$ by restricting the cubes R to those contained in Q.

Remark that if $p(x)=1$ for some $x \in {\mathbb{R}}^{n}$, then we define $\sigma (x)=1$.

If $p(\cdot )\equiv p$ is a constant exponent, then $A_{p(\cdot )}^{\mathrm{loc}}$ coincides with the class $A_{p}^{\mathrm{loc}}$ defined in [14]. Using a different method, we seek to establish that the local analog of the result in [2, 3] is available: Let f be a measurable function and let $M^{\mathrm{loc}}$ be the local maximal operator given by

$$ M^{\mathrm{loc}}f(x) \equiv \sup_{Q \in \mathcal{Q}, \vert Q \vert \le 1} \frac{\chi _{Q}(x)}{ \vert Q \vert } \int _{Q} \bigl\vert f(y) \bigr\vert {\,\mathrm{d}}y \quad \bigl(x \in { \mathbb{R}}^{n}\bigr). $$

Needless to say, this is an analog of the Hardy–Littlewood maximal operator given by

$$ Mf(x) \equiv \sup_{Q \in \mathcal{Q}} \frac{\chi _{Q}(x)}{ \vert Q \vert } \int _{Q} \bigl\vert f(y) \bigr\vert {\,\mathrm{d}}y \quad \bigl(x \in {\mathbb{R}}^{n}\bigr). $$

For the boundedness of M, we postulate the following two conditions on $p(\cdot )$:

• (1) The local log-Hölder continuity condition, which is given by

  $$ {\mathrm{LH}}_{0}: \bigl\vert p(x)-p(y) \bigr\vert \le \frac{C}{-\log \vert x-y \vert }, \quad x,y \in \mathbb{R}^{n}, \vert x-y \vert \le \frac{1}{2}. $$

  (1.1)

• (2) The log-Hölder continuity condition at infinity. That is, there exists $p_{\infty }\in [0,\infty )$ such that

  $$ {\mathrm{LH}}_{\infty }: \bigl\vert p(x)-p_{\infty } \bigr\vert \le \frac{C}{\log (e+ \vert x \vert )}, \quad x \in \mathbb{R}^{n}. $$

  (1.2)

Keeping these in mind, we state the main result of this paper.

Theorem 1.2

Let $p(\cdot ):\mathbb{R}^{n} \to [1,\infty )$ satisfy conditions (1.1) and (1.2) and $1< p_{-} \equiv \operatorname{essinf}_{x \in {\mathbb{R}}^{n}} p(x) \le p_{+} \equiv \operatorname{esssup}_{x \in {\mathbb{R}}^{n}} p(x) <\infty$. For any given $w \in A_{p(\cdot )}^{\mathrm{loc}}$, there exists a constant $C>0$ such that for all measurable functions f,

$$ \bigl\Vert M^{\mathrm{loc}}f \bigr\Vert _{L^{p(\cdot )}(w)} \le C \Vert f \Vert _{L^{p(\cdot )}(w)}. $$

It is easy to show that $w \in A_{p(\cdot )}^{\mathrm{loc}}$ is necessary for the boundedness of $M^{\mathrm{loc}}$, since $M^{\mathrm{loc}}f(x) \ge \frac{1}{|Q|}\int _{Q}|f(y)|{\,\mathrm{d}}x$ for all cubes Q with a volume less than or equal to 1 containing x.

Additionally, the matters are reduced to the estimate of the following maximal function. We consider the local maximal operator given by

$$ M^{\mathrm{loc}}_{6^{-1}}f(x) \equiv \sup_{Q \in \mathcal{Q}, \vert Q \vert \le 6^{-n}} \frac{\chi _{Q}(x)}{ \vert Q \vert } \int _{Q} \bigl\vert f(y) \bigr\vert {\,\mathrm{d}}y \quad \bigl(x \in { \mathbb{R}}^{n}\bigr), $$

for a measurable function f. In fact, if we denote the seven-fold composition of $M^{\mathrm{loc}}_{6^{-1}}$ by $(M^{\mathrm{loc}}_{6^{-1}})^{7}$, then there exists a constant $C>0$ such that $M^{\mathrm{loc}}f \le C(M^{\mathrm{loc}}_{6^{-1}})^{7}f$ for any measurable function f.

Before we go further, we offer some words on the technique of the proof. At first glance, the proof of Theorem 1.2 seems to be a reexamination of the original theorem [3, Theorem 1.5], which is recalled below.

Proposition 1.3

([3, Theorem 1.5])

Suppose that we have a variable exponent $p(\cdot ):\mathbb{R}^{n}\to [1,\infty )$ such that $1\le p_{-}\le p_{+}<\infty$ and that $p(\cdot )$ satisfies (1.1) and (1.2). Let $w \in A_{p(\cdot )}$. Then, the Hardy–Littlewood maximal operator M satisfies the weak-type inequality

$$ \Vert t\chi _{\{x \in \mathbb{R}^{n}:Mf(x)>t\}}w \Vert _{p( \cdot )} \le C \Vert fw \Vert _{p(\cdot )}, \quad t>0 $$

for all measurable functions f. If, in addition, $p_{-}>1$, then

$$ \bigl\Vert (Mf)w \bigr\Vert _{p(\cdot )} \le C \Vert fw \Vert _{p(\cdot )} $$

for all measurable functions f.

However, as the example of $w(x)=\exp (|x|) \in A_{p(\cdot )}^{\mathrm{loc}} \setminus A_{p(\cdot )}$ shows, inequality

$$ \int _{{\mathbb{R}}^{n}}\frac{w(x)^{p(x)}\,\mathrm{d}x}{(e+ \vert x \vert )^{K}}, \int _{{ \mathbb{R}}^{n}}\frac{\sigma (x)\,\mathrm{d}x}{(e+ \vert x \vert )^{K}}< \infty , $$

which is used in the proof of [3, Theorem 1.5], fails for local Muckenhoupt class with variable exponents. Hence, the proof of [3] cannot be used naively for the local Muckenhoupt class. This observation led to the technique of Rychkov, who detailed a method for creating global weights from given local weights.

Next, we consider why the technique employed by Rychkov [14] does not work directly. For simplicity, we work in ${\mathbb{R}}$. In [14], Rychkov considered a symmetric extension of weights. More precisely, given an interval I and a weight w on I, Rychkov defined a weight $w_{I}$ on an interval J adjacent to I mirror-symmetrically with respect to the contact point in $I \cap J$. Here, we tried to repeat this procedure to define a weight $w_{I}$ on ${\mathbb{R}}$. Hence, this method is not applicable because we cannot extend the variable exponents mirror-symmetrically. For example, if the weight w satisfies $w(t)=|t|^{-\frac{1}{3}}$ on $(-2,2)$ and the exponent $p(\cdot )$ satisfies $p(t)=2$ on $(-1,1)$ and $p(t)=\frac{4}{3}$ on $(3,5)$, then the weight $w_{(-2,2)}$ defined mirror-symmetrically from $w|_{(-2,2)}$ does not satisfy $\sigma =w_{(-2,2)}{}^{-\frac{1}{p(\cdot )-1}} \in L^{1}_{\mathrm{loc}}(2,6)$.

To overcome these issues, two devices are necessary. The first device is well known. We fix a dyadic grid ${\mathcal{D}}_{k,{\mathbf{a}}}$, $k \in {\mathbb{Z}}$ and ${\mathbf{a}} \in \{0,1,2\}^{n}$. More precisely, we let

$$ {\mathcal{D}}_{k,a}^{0} \equiv \bigl\{ 2^{-k}[m+a/3,m+a/3+1):m \in { \mathbb{Z}}\bigr\} $$

for $k \in {\mathbb{Z}}$ and $a=0,1,2$, and consider

$$ {\mathcal{D}}_{k,{\mathbf{a}}} \equiv \bigl\{ Q_{1} \times Q_{2} \times \cdots \times Q_{n}: Q_{j} \in {\mathcal{D}}_{k,a_{j}}^{0}, j=1,2, \ldots ,n \bigr\} $$

for $k \in {\mathbb{Z}}$ and ${\mathbf{a}}=(a_{1},a_{2},\ldots ,a_{n}) \in \{0,1,2\}^{n}$. See [4, Lemma 4.8] for this construction. Herein, a dyadic grid is the family $\mathcal{D}_{\mathbf{a}} \equiv \bigcup_{k \in {\mathbb{Z}}}{ \mathcal{D}}_{k,{\mathbf{a}}}$ for ${\mathbf{a}} \in \{0,1,2\}^{n}$. It is noteworthy that for any cube Q there exists $R \in \bigcup_{{\mathbf{a}} \in \{0,1,2\}^{n}}{\mathcal{D}}_{\mathbf{a}}$ such that $Q \subset R$ and $|R| \le 6^{n}|Q|$. As in [11], we reduced the matters to the local maximal operator generated by a family ${\mathcal{D}}$ given by

$$ M^{\mathrm{loc}}_{\mathcal{D}}f(x) \equiv \sup_{Q \in {\mathcal{D}}, \vert Q \vert \le 1} \frac{\chi _{Q}(x)}{ \vert Q \vert } \int _{Q} \bigl\vert f(y) \bigr\vert {\,\mathrm{d}}y \quad \bigl(x \in { \mathbb{R}}^{n}\bigr) $$

for a measurable function f and a dyadic grid ${\mathcal{D}} \in \{{\mathcal{D}}_{{\mathbf{a}}}: {\mathbf{a}} \in \{0,1,2\}^{n} \}$. In fact, we have

$$ M^{\mathrm{loc}}_{6^{-1}}f(x)\le C \sum_{{\mathbf{a}} \in \{0,1,2\}^{n}} M^{ \mathrm{loc}}_{{\mathcal{D}}_{\mathbf{a}}}f(x) \quad \bigl(x \in \mathbb{R}^{n} \bigr). $$

Here and below, due to the similarity, we suppose ${\mathbf{a}}=(1,1,\ldots ,1)$. We abbreviate ${\mathcal{D}}_{(1,1,\ldots ,1)}$ to ${\mathfrak{D}}$. The other values of a can be handled similarly.

Since we reduced the matters to a dyadic grid ${\mathfrak{D}}={\mathcal{D}}_{(1,1,\ldots ,1)}$, it is natural to define the class $A_{p(\cdot )}^{\mathrm{loc}}({\mathfrak{D}})$.

Definition 1.4

Given an exponent $p(\cdot ):\mathbb{R}^{n} \to (1,\infty )$ with $p_{-}>1$ and a weight w, we say that $w \in A^{\mathrm{loc}}_{p(\cdot )}({\mathfrak{D}})$ if

$$ [w]_{A^{\mathrm{loc}}_{p(\cdot )}({\mathfrak{D}})} \equiv \sup_{Q \in { \mathfrak{D}}, \vert Q \vert \le 1} \vert Q \vert ^{-1} \Vert \chi _{Q} \Vert _{L^{p(\cdot )}(w)} \Vert \chi _{Q} \Vert _{L^{p'(\cdot )}(\sigma )}< \infty , $$

where $\sigma \equiv w^{-\frac{1}{p(\cdot )-1}}$ and the supremum is taken over all cubes $Q \in {\mathfrak{D}}$ with $|Q|\le 1$.

In an analogy to Theorem 1.2, we can prove the following theorem:

Theorem 1.5

Let $p(\cdot ):\mathbb{R}^{n} \to [1,\infty )$ satisfy conditions (1.1) and (1.2). If $1< p_{-} \le p_{+} <\infty$, then, for any $w \in A_{p(\cdot )}^{\mathrm{loc}}({\mathfrak{D}})$, there exists a constant $C>0$ such that

$$ \bigl\Vert M^{\mathrm{loc}}_{{\mathfrak{D}}}f \bigr\Vert _{L^{p(\cdot )}(w)} \le C \Vert f \Vert _{L^{p( \cdot )}(w)} $$

for any $f \in L^{p(\cdot )}(w)$.

The second device is a new local/global strategy. To prove Theorem 1.5 when dealing with local dyadic Muckenhoupt weights, we consider (global) dyadic Muckenhoupt weights.

Definition 1.6

Given an exponent $p(\cdot ):\mathbb{R}^{n} \to [1,\infty )$ with $p_{-}>1$ and a weight w, we say that $w \in A_{p(\cdot )}({\mathfrak{D}})$ if

$$ [w]_{A_{p(\cdot )}({\mathfrak{D}})} \equiv \sup_{Q \in {\mathfrak{D}}} \vert Q \vert ^{-1} \Vert \chi _{Q} \Vert _{L^{p(\cdot )}(w)} \Vert \chi _{Q} \Vert _{L^{p'(\cdot )}(\sigma )}< \infty , $$

where $p'(\cdot )$ is the conjugate exponent of $p(\cdot )$, $\sigma \equiv w^{-\frac{1}{p(\cdot )-1}}$ and the supremum is taken over all cubes $Q \in {\mathfrak{D}}$.

Here, we propose a new method to create a globally regular weight $w_{Q} \in A_{p(\cdot )}({\mathfrak{D}})$, $Q \in {\mathfrak{D}}$ from a weight $w \in A_{p(\cdot )}^{\mathrm{loc}}({\mathfrak{D}})$ in Proposition 2.10. As we will see, this technique is valid only for the dyadic maximal operator (see Remark 2.11). In addition to the local/global strategy which differs from that in Rychkov [14], we use the localization principle due to Hästo [8, Theorem 2.4]. In analogy to Theorem 1.2, we can prove the following theorem:

Theorem 1.7

Let $p(\cdot ):\mathbb{R}^{n} \to [1,\infty )$ satisfy conditions (1.1) and (1.2). If $1< p_{-} \le p_{+} <\infty$, then given any $w \in A_{p(\cdot )}({\mathfrak{D}})$, there exists a constant $C>0$ such that

$$ \Vert M_{{\mathfrak{D}}}f \Vert _{L^{p(\cdot )}(w)} \le C \Vert f \Vert _{L^{p(\cdot )}(w)} $$

for any measurable function f.

As explained above, Theorem 1.2 will be proven once we prove Theorem 1.5, whose proof uses Theorem 1.7. We note that unlike the proof of Theorem 1.2, that for Theorem 1.7 is an analog of [3, Theorem 1.5]. However, we include its whole proof for the sake of completeness.

Finally, as an application of our results, we will prove the Rubio de Francia extrapolation theorem in our setting of weights, which in turn produces the weighted vector-valued maximal inequality. The theory of extrapolation is a powerful tool in harmonic analysis to extend many results starting from a weighted inequality. Cruz-Uribe and Wang [5] and Ho [9] extended the extrapolation theorem to weighted Lebesgue spaces with variable exponents. We will show the extrapolation theorem for $A^{\mathrm{loc}}_{p(\cdot )}$ by applying the boundedness of the local maximal operator.

Theorem 1.8

Let $N:[1,\infty )\to [1,\infty )$ be an increasing function. Suppose that for some $p_{0}, 1< p_{0}<\infty$, and every $w_{0} \in A^{\mathrm{loc}}_{p_{0}}$, the inequality

$$ \int _{\mathbb{R}^{n}} f(x)^{p_{0}} w_{0}(x) { \,\mathrm{d}}x \le N\bigl([w_{0}]_{A^{ \mathrm{loc}}_{p_{0}}}\bigr) \int _{\mathbb{R}^{n}} g(x)^{p_{0}} w_{0}(x) { \,\mathrm{d}}x $$

(1.3)

holds for pairs of functions $(f,g)$ contained in some family $\mathcal{F}$ of nonnegative measurable functions. Let $p(\cdot )$ satisfy conditions (1.1) and (1.2) as well as $1< p_{-}\le p_{+}<\infty$. Also let $w \in A^{\mathrm{loc}}_{p(\cdot )}$. Then,

$$ \Vert f \Vert _{L^{p(\cdot )}(w)} \le C \Vert g \Vert _{L^{p(\cdot )}(w)} $$

for $(f, g) \in \mathcal{F}$.

Rychkov [14, Lemma 2.11] proved the weighted vector-valued inequality for $M^{\mathrm{loc}}$ and $w \in A^{\mathrm{loc}}_{p}$ as an extension of the results in [1].

Proposition 1.9

Let $1< p<\infty$, $1< q \le \infty$, and $w \in A^{\mathrm{loc}}_{p}$. Then for any sequence of measurable functions $\{f_{j}\}_{j \in \mathbb{N}}$, we have

$$ \Biggl\Vert \Biggl( \sum_{j=1}^{\infty } \bigl[ M^{\mathrm{loc}}{}f_{j} \bigr]^{q} \Biggr)^{\frac{1}{q}} \Biggr\Vert _{L^{p}(w)} \le C \Biggl\Vert \Biggl( \sum_{j=1}^{\infty } \vert f_{j} \vert ^{q} \Biggr)^{\frac{1}{q}} \Biggr\Vert _{L^{p}(w)}. $$

Recall that Cruz-Uribe et al. extended the same result by Anderson and John [1] to variable Lebesgue spaces.

Proposition 1.10

Suppose that $p(\cdot )$ satisfies conditions (1.1) and (1.2), as well as $1< p_{-}\le p_{+}<\infty$, and let $w \in A_{p(\cdot )}$ and $1< q \le \infty$. Then for any sequence of measurable functions $\{f_{j}\}_{j \in \mathbb{N}}$, we have

$$\begin{aligned} \Biggl\Vert \Biggl( \sum_{k=1}^{\infty } [M f_{k}{} ]^{q} \Biggr)^{\frac{1}{q}} \Biggr\Vert _{L^{p(\cdot )}(w)} \le C \Biggl\Vert \Biggl( \sum_{k=1}^{\infty } \vert f_{k} \vert ^{q} \Biggr)^{\frac{1}{q}} \Biggr\Vert _{L^{p(\cdot )}(w)}. \end{aligned}$$

(1.4)

The following theorem is the weighted vector-valued inequality for the local variable weight:

Theorem 1.11

Suppose that $p(\cdot )$ satisfies conditions (1.1) and (1.2), as well as $1< p_{-}\le p_{+}<\infty$, and let $w \in A_{p(\cdot )}^{\mathrm{loc}}$ and $1< q \le \infty$. Then for any sequence of measurable functions $\{f_{j}\}_{j \in \mathbb{N}}$, we have

$$\begin{aligned} \Biggl\Vert \Biggl( \sum_{k=1}^{\infty } \bigl[M^{\mathrm{loc}}f_{k}{} \bigr]^{q} \Biggr)^{\frac{1}{q}} \Biggr\Vert _{L^{p(\cdot )}(w)} \le C \Biggl\Vert \Biggl( \sum_{k=1}^{\infty } \vert f_{k} \vert ^{q} \Biggr)^{\frac{1}{q}} \Biggr\Vert _{L^{p(\cdot )}(w)}. \end{aligned}$$

(1.5)

Throughout the paper, we use the following notation. The relation $A \lesssim B$ means $A \le CB$ for some constant $C>0$, while $A\gtrsim B$ means $A \ge CB$ for some constant $C>0$. The relation $A \sim B$ means that $A\lesssim B$ and $B\lesssim A$. For a weight w and measurable set E, we define $w(E)\equiv \int _{E} w(x){\,\mathrm{d}}x$.

The rest of this paper is organized as follows. Sect. 2 establishes various preliminaries and some notation. Sect. 3 proves Theorem 1.7, while Sect. 4 proves Theorem 1.5, which includes Theorem 1.2. Finally, as an application, Sect. 5 is devoted to the proof of the weighted vector-valued maximal inequality for $A^{\mathrm{loc}}_{p(\cdot )}$.

2 Preliminaries

We collect some preliminary facts by investigating some elementary properties of the dyadic grids ${\mathcal{D}}_{k,1}^{0}$ and ${\mathcal{D}}_{k,(1,1,\ldots ,1)}$ and recalling the definition of variable Lebesgue spaces. Then we consider classes of weights.

2.1 Dyadic grids ${\mathcal{D}}_{k,1}^{0}$ and ${\mathcal{D}}_{k,(1,1,\ldots ,1)}$

Recall that the grid ${\mathcal{D}}_{k,1}^{0}$ is given by

$$ {\mathcal{D}}_{k,1}^{0} \equiv \bigl\{ [2^{-k}(m+1/3),2^{-k}(m+4/3)):m \in {\mathbb{Z}} \bigr\} \quad (k \in {\mathbb{Z}}). $$

Thus, $0 \in V_{k}\equiv [-2^{1-k}/3,2^{-k}/3)\in {\mathcal{D}}_{k,1}^{0}$ for all $k \in {\mathbb{Z}}$. Concerning ${\mathcal{D}}_{k,(1,1,\ldots ,1)}$, we employ the following property.

Lemma 2.1

Suppose that $Q \in {\mathcal{D}}_{(1,1,\ldots ,1)}$ does not contain 0. If we set

$$ U_{k}=\underbrace{V_{k} \times \cdots \times V_{k}}_{n \textit{ times}}, \quad k_{Q}=\max \{k \in { \mathbb{Z}}:Q \cap U_{k} \ne \emptyset \}, $$

then $Q \subset U_{k_{Q}-1}$ and $Q \cap U_{k_{Q}+1}=\emptyset$. In particular, $\ell (Q) \sim 2^{-k_{Q}}$.

Proof

By the property of “max”, it is clear that $Q \cap U_{k_{Q}+1}=\emptyset$. Let us prove $Q \subset U_{k_{Q}-1}$. First, we prove Q does not contain $U_{k_{Q}}$. To this end, it suffices prove $\ell (Q)\le \ell (U_{k_{Q}})$. Since we can concentrate on the $x_{1}$-direction, we may assume $n=1$. Write

$$ Q=[2^{-k}(m+1/3),2^{-k}(m+4/3)) $$

with $k,m \in {\mathbb{Z}}$. Since $0 \notin Q$, $m \ne -1$. Since $U_{k_{Q}}$ and Q meet at a point, a geometric observation shows

$$ \begin{gathered}2^{-k} \biggl(m+\frac{1}{3} \biggr)< \frac{2^{-k_{Q}}}{3} \le 2^{-k} \biggl(m+ \frac{4}{3} \biggr) \quad \text{or}\\ 2^{-k} \biggl(m+\frac{1}{3} \biggr) \le \frac{-2^{1-k_{Q}}}{3}< 2^{-k} \biggl(m+ \frac{4}{3} \biggr). \end{gathered} $$

(2.1)

Equivalently,

$$ 3m+1< 2^{k-k_{Q}} \le 3m+4 \quad \text{or}\quad -(3m+4)< 2^{k-k_{Q}} \le -(3m+1). $$

Thus, since $2^{k-k_{Q}}>0$, $m \ge 0$ or $m \le -2$, in which case $k \ge k_{Q}$.

Next, we will show that $Q \subset U_{k_{Q}-1}$. If $Q \subset U_{k_{Q}}$, this is clear. We assume otherwise. Then, the relations (2.1) hold. Since the left relation in (2.1) holds, we have

$$\begin{aligned} \frac{2^{-(k_{Q}-1)}}{3} - \biggl( 2^{-k} \biggl(m+ \frac{4}{3} \biggr) \biggr) = \frac{2^{-k_{Q}}}{3} + \biggl( \frac{2^{-k_{Q}}}{3}- 2^{-k} \biggl(m+\frac{1}{3} \biggr) - \frac{2^{-k}}{3} \biggr) >0. \end{aligned}$$

Similarly, since the right relation in (2.1) holds, we have

$$ 2^{-k} \biggl(m+\frac{1}{3} \biggr) - \frac{-2^{1-(k_{Q}-1)}}{3}>0. $$

Therefore, we see that $Q \subset U_{k_{Q}-1}$. □

2.2 Weighted variable Lebesgue spaces

For any measurable subset $\Omega \subset \mathbb{R}^{n}$, denote

$$ p_{+}(\Omega ) \equiv \mathop{\operatorname{esssup}}\limits_{x \in \Omega } p(x), \qquad p_{-}( \Omega ) \equiv \mathop{\operatorname{essinf}}\limits_{x \in \Omega } p(x). $$

In particular, when $\Omega =\mathbb{R}^{n}$, we simply write $p_{+}$ and $p_{-}$, respectively.

Let $1< p(\cdot ) <\infty$. If $p(\cdot ) \in {\mathrm{LH}}_{0}$ then $p'(\cdot ) \in {\mathrm{LH}}_{0}$. Likewise, if $p(\cdot ) \in {\mathrm{LH}}_{\infty }$ then $p'(\cdot ) \in {\mathrm{LH}}_{\infty }$. Furthermore, $(p_{\infty })'=(p')_{\infty }$.

In addition, recall generalized Hölder’s inequality.

Lemma 2.2

(Generalized Hölder’s inequality)

Let $p(\cdot ) :{\mathbb{R}}^{n} \to [1,\infty ]$ be a variable exponent. Then for all $f\in L^{p(\cdot )}({\mathbb{R}}^{n})$ and all $g\in L^{p'(\cdot )}({\mathbb{R}}^{n})$,

$$ \Vert f \cdot g \Vert _{L^{1}({\mathbb{R}}^{n})} \le r_{p} \Vert f \Vert _{L^{p(\cdot )}({ \mathbb{R}}^{n})} \Vert g \Vert _{L^{p'(\cdot )}({\mathbb{R}}^{n})}, $$

(2.2)

where

$$ r_{p}\equiv 1+\frac{1}{p_{-}} - \frac{1}{p_{+}}= \frac{1}{p_{-}}+ \frac{1}{(p')_{-}} \le 2. $$

(2.3)

Let us recall some properties for the variable Lebesgue space $L^{p(\cdot )}({\mathbb{R}}^{n})$.

Lemma 2.3

([12, Lemma 2.2])

Suppose that $p(\cdot )$ is a function satisfying (1.1) and (1.2).

• (1) For all cubes Q with $|Q| \le 1$, we have $|Q|^{{1}/{p_{-}(Q)}} \lesssim |Q|^{{1}/{p_{+}(Q)}}$. In particular, we have $|Q|^{{1}/{p_{-}(Q)}} \sim |Q|^{{1}/{p_{+}(Q)}} \sim |Q|^{{1}/{p(z)}} \sim \|\chi _{Q}\|_{L^{p(\cdot )}}$.

• (2) For all cubes Q with $|Q| \ge 1$, we have $\|\chi _{Q}\|_{L^{p(\cdot )}} \sim |Q|^{{1}/{p_{\infty }}}$.

Lemma 2.4

([3, Lemma 2.2], [13, Lemma 2.17])

Let $p(\cdot ):\mathbb{R}^{n} \to [1,\infty )$ be such that $p_{+}<\infty$. Then given any set Ω and any measurable function f,

• (1) if $\|f\chi _{\Omega }\|_{p(\cdot )} \le 1$, then $\|f\chi _{\Omega }\|_{p(\cdot )}^{p_{+}(\Omega )} \le \int _{ \Omega } |f(x)|^{p(x)} {\,\mathrm{d}}x \le \|f\chi _{\Omega }\|_{p(\cdot )}^{p_{-}( \Omega )}$ and

  $$ \int _{\Omega } \bigl\vert f(x) \bigr\vert ^{p(x)} { \,\mathrm{d}}x \le \Vert f \Vert _{p(\cdot )}, $$

• (2) if $\|f\chi _{\Omega }\|_{p(\cdot )} \ge 1$, then $\|f\chi _{\Omega }\|_{p(\cdot )}^{p_{-}(\Omega )} \le \int _{ \Omega } |f(x)|^{p(x)} {\,\mathrm{d}}x \le \|f\chi _{\Omega }\|_{p(\cdot )}^{p_{+}( \Omega )}$.

Lemma 2.5

Let $p(\cdot ) :\mathbb{R}^{n} \to [1, \infty )$ and f be a measurable function. Then $\|f\|_{p(\cdot )} \le 1$ if and only if $\int _{\mathbb{R}^{n}} |f(x)|^{p(x)} {\,\mathrm{d}}x \le 1$.

Remark 2.6

Let Q be a cube. In Lemmas 2.4 and 2.5, let $f=w^{\frac{1}{p(\cdot )}}\chi _{Q}$ to obtain the following equivalence:

$$ \Vert \chi _{Q} \Vert _{L^{p(\cdot )}(w)} \le 1 \quad \Longleftrightarrow\quad \int _{Q} w(x) {\,\mathrm{d}}x \le 1. $$

(2.4)

A direct consequence of (2.4) is the following:

• (1) If $\|\chi _{Q}\|_{L^{p(\cdot )}(w)}\le 1$, then

  $$ \Vert \chi _{Q} \Vert _{L^{p(\cdot )}(w)}^{p_{+}(Q)} \le w(Q) \le \Vert \chi _{Q} \Vert _{L^{p(\cdot )}(w)}^{p_{-}(Q)}. $$

  (2.5)

• (2) If $\|\chi _{Q}\|_{L^{p(\cdot )}(w)}\ge 1$, then

  $$ \Vert \chi _{Q} \Vert _{L^{p(\cdot )}(w)}^{p_{-}(Q)} \le w(Q) \le \Vert \chi _{Q} \Vert _{L^{p(\cdot )}(w)}^{p_{+}(Q)}. $$

  (2.6)

The following inequality is a key tool used in this paper. Although [3, Lemma 2.7] considers Borel measures, we consider Lebesgue measures. Here and below denote by $L^{0}({\mathbb{R}}^{n})$ the set of all measurable functions defined over ${\mathbb{R}}^{n}$.

Lemma 2.7

([3, Lemma 2.7])

Let μ be a weighted Lebesgue measure defined on a measurable set G. Given a set G and two exponents $s(\cdot )$ and $r(\cdot )$ such that

$$ \bigl\vert s(y)-r(y) \bigr\vert \lesssim \frac{1}{\log (e+ \vert y \vert )} \quad (y \in G). $$

Then for all $t \in [1,\infty )$ and $f \in L^{0}({\mathbb{R}}^{n})$ with $0 \le f \le 1$,

$$ \int _{G} f(y)^{s(y)}\,\mathrm{d}\mu (y) \lesssim \int _{G} f(y)^{r(y)}\,\mathrm{d}\mu (y) + \int _{G} \frac{\mathrm{d}\mu (y)}{(e+ \vert y \vert )^{t n r_{-}(G)}}. $$

Finally, recall the localization principle due to Hästo.

Lemma 2.8

([8])

Let $p(\cdot ):\mathbb{R}^{n} \to [1,\infty )$ satisfy conditions (1.1) and (1.2). Then

$$ \Vert f \Vert _{L^{p(\cdot )}} \sim \biggl( \sum _{Q \in {\mathcal{D}}_{0,(1,1, \ldots ,1)}} \bigl( \Vert f\chi _{Q} \Vert _{L^{p(\cdot )}}\bigr)^{p_{\infty }} \biggr)^{ \frac{1}{p_{\infty }}} $$

for all measurable functions f.

2.3 Weights

Here and below, we assume that $p(\cdot )$ takes value in $(1, \infty )$ and satisfies conditions (1.1) and (1.2). First, note that for $w \in A_{p(\cdot )}^{\mathrm{loc}}$ we have, by the definition of $A_{p(\cdot )}^{\mathrm{loc}}$,

$$ \bigl\Vert \bigl([w]_{A_{p(\cdot )}^{\mathrm{loc}}} \vert Q \vert \bigr)^{-1} \bigr\Vert \chi _{Q} \|_{L^{p'(\cdot )}(\sigma )}\chi _{Q} \| _{L^{p(\cdot )}(w)} \le 1, $$

or equivalently,

$$ \int _{Q} \biggl( \frac{ \Vert \chi _{Q} \Vert _{L^{p'(\cdot )}(\sigma )}}{[w]_{A_{p(\cdot )}^{\mathrm{loc}}} \vert Q \vert } \biggr)^{p(x)} w(x)\,\mathrm{d}x \le 1. $$

(2.7)

for $Q \in {\mathcal{Q}}$ with $|Q| \le 1$.

Recall an equivalent definition of $A_{\infty }$. We refer to [7, Theorem 7.3.3] for its proof.

Lemma 2.9

([7, Theorem 7.3.3])

Given a weight w, the following are equivalent:

• (1) $w \in A_{\infty }$.

• (2) There exist constants $0< C_{1}$, $C_{2}<1$ such that given any cube Q and any measurable set $E \subset Q$ with $|E|>C_{1}|Q|$, then $w(E)>C_{2}w(Q)$.

If $(2)$ holds, then it can be arranged so that $C_{1}$ and $C_{2}$ depend only on the $A_{\infty }$ constant of w.

The next lemma is important in this paper. Rychkov extended a local weight mirror-symmetrically but a variable exponent cannot be set in this manner. Hence, we propose a different extension.

Proposition 2.10

Suppose that $p_{-}>1$. Let $w \in A^{\mathrm{loc}}_{p(\cdot )}({\mathfrak{D}})$. Let $I \in {\mathfrak{D}}$ be a cube with $|I|=1$. Define

$$ \overline{w}(x) \equiv \textstyle\begin{cases} ( \Vert \chi _{I} \Vert _{L^{p(\cdot )}(w)})^{p(x)}&(x \in {\mathbb{R}}^{n} \setminus I), \\ w(x)&(x \in I). \end{cases} $$

Then $\overline{w} \in A_{p(\cdot )}({\mathfrak{D}})$ and $[\overline{w}]_{A_{p(\cdot )}({\mathfrak{D}})} \lesssim [w]_{A^{\mathrm{loc}}_{p( \cdot )}({\mathfrak{D}})}$.

Proof

Arithmetic shows that $w \in A^{\mathrm{loc}}_{p(\cdot )}({\mathfrak{D}})$ if and only if $\sigma \in A^{\mathrm{loc}}_{p'(\cdot )}({\mathfrak{D}})$. We also note that $[w]_{A_{p(\cdot )}^{\mathrm{loc}}({\mathfrak{D}})} \gtrsim 1$ due to Hölder’s inequality (see Lemma 2.2). Write $\overline{\sigma }\equiv \overline{w}^{-\frac{1}{p(\cdot )-1}}$. Let $Q \in {\mathfrak{D}}$. We need to estimate

$$ \frac{1}{ \vert Q \vert } \Vert \chi _{Q} \Vert _{L^{p(\cdot )}(\overline{w})} \Vert \chi _{Q} \Vert _{L^{p'( \cdot )}(\overline{\sigma })}. $$

We distinguish three cases:

• Suppose $I \cap Q=\emptyset$. In this case, by virtue of Lemma 2.3,

  $$ \frac{1}{ \vert Q \vert } \Vert \chi _{Q} \Vert _{L^{p(\cdot )}(\overline{w})} \Vert \chi _{Q} \Vert _{L^{p'( \cdot )}(\overline{\sigma })} = \frac{1}{ \vert Q \vert } \Vert \chi _{Q} \Vert _{L^{p( \cdot )}} \Vert \chi _{Q} \Vert _{L^{p'(\cdot )}} \sim 1 \le [w]_{A_{p(\cdot )}^{ \mathrm{loc}}({\mathfrak{D}})}. $$

• Next, suppose $Q \subset I$. In this case, since $w \in A^{\mathrm{loc}}_{p(\cdot )}({\mathfrak{D}})$,

  $$ \frac{1}{ \vert Q \vert } \Vert \chi _{Q} \Vert _{L^{p(\cdot )}(\overline{w})} \Vert \chi _{Q} \Vert _{L^{p'( \cdot )}(\overline{\sigma })} = \frac{1}{ \vert Q \vert } \Vert \chi _{Q} \Vert _{L^{p( \cdot )}(w)} \Vert \chi _{Q} \Vert _{L^{p'(\cdot )}(\sigma )} \le [w]_{A^{\mathrm{loc}}_{p( \cdot )}({\mathfrak{D}})}. $$

• Finally, suppose $Q \supset I$. In this case, again by virtue of Lemma 2.3 and the fact that $w \in A^{\mathrm{loc}}_{p(\cdot )}({\mathfrak{D}})$,

  $$\begin{aligned} &\frac{1}{ \vert Q \vert } \Vert \chi _{Q} \Vert _{L^{p(\cdot )}(\overline{w})} \Vert \chi _{Q} \Vert _{L^{p'(\cdot )}(\overline{\sigma })} \\ &\quad \sim \frac{1}{ \vert Q \vert } \bigl( \Vert \chi _{Q \setminus I} \Vert _{L^{p(\cdot )}( \overline{w})}+ \Vert \chi _{I} \Vert _{L^{p(\cdot )}(\overline{w})} \bigr) \bigl( \Vert \chi _{Q \setminus I} \Vert _{L^{p'(\cdot )}(\overline{\sigma })}+ \Vert \chi _{I} \Vert _{L^{p'(\cdot )}(\overline{\sigma })} \bigr) \\ &\quad \sim \frac{1}{ \vert Q \vert } \bigl( \vert Q \vert ^{\frac{1}{p_{\infty }}} \Vert \chi _{I} \Vert _{L^{p( \cdot )}(w)}+ \Vert \chi _{I} \Vert _{L^{p(\cdot )}(w)} \bigr) \biggl( \frac{ \vert Q \vert ^{\frac{1}{p_{\infty }'}}}{ \Vert \chi _{I} \Vert _{L^{p(\cdot )}(w)}} + \Vert \chi _{I} \Vert _{L^{p'(\cdot )}(\sigma )} \biggr) \\ &\quad \sim \frac{1}{ \vert Q \vert } \vert Q \vert ^{\frac{1}{p_{\infty }}} \bigl( \vert Q \vert ^{ \frac{1}{p_{\infty }'}} + \Vert \chi _{I} \Vert _{L^{p(\cdot )}(w)} \Vert \chi _{I} \Vert _{L^{p'( \cdot )}(\sigma )} \bigr) \\ &\quad \lesssim [w]_{A^{\mathrm{loc}}_{p(\cdot )}({\mathfrak{D}})}, \end{aligned}$$

  as required.

 □

Remark 2.11

Let $I\equiv (0,1)$ and $J\equiv (-1,0)$. Define $w(t)=t^{-\frac{1}{2}}$ on I and $w(t)=1$ on J. Denote by $A_{2}(I)$ the $A_{2}$ class over I. Although $w \in A_{2}(I)$, w is not in $A_{2}(I \cup J \cup \{0\})$.

Lemma 2.12

Suppose $w \in A_{p(\cdot )}({\mathfrak{D}})$. Then for all $Q \in {\mathfrak{D}}$ and a measurable subset E of Q,

$$ \frac{ \vert E \vert }{ \vert Q \vert } \le 2[w]_{A_{p(\cdot )}({\mathfrak{D}})} \frac{ \Vert \chi _{E} \Vert _{L^{p(\cdot )}(w)}}{ \Vert \chi _{Q} \Vert _{L^{p(\cdot )}(w)}}. $$

Here for the sake of convenience, we include the proof.

Proof

By Hölder’s inequality (see Lemma 2.2), we have

$$ \vert E \vert = \int _{E} \mathrm{d}x \le 2 \Vert \chi _{E} \Vert _{L^{p(\cdot )}(w)} \Vert \chi _{Q} \Vert _{L^{p'( \cdot )}(\sigma )} \le 2[w]_{A_{p(\cdot )}({\mathfrak{D}})} \vert Q \vert \frac{ \Vert \chi _{E} \Vert _{L^{p(\cdot )}(w)}}{ \Vert \chi _{Q} \Vert _{L^{p(\cdot )}(w)}}, $$

as required. □

Lemma 2.13

Suppose $w \in A_{p(\cdot )}({\mathfrak{D}})$. Then for all $Q \in {\mathfrak{D}}$,

$$ \Vert \chi _{Q} \Vert _{L^{p(\cdot )}(w)}^{p_{-}(Q)-p_{+}(Q)} \lesssim 1. $$

Proof

We assume $\|\chi _{Q}\|_{L^{p(\cdot )}(w)} \le 1$. Otherwise the inequality is trivial since $p_{-}(Q) \le p_{+}(Q)$. We distinguish 3 cases. Let $Q_{0}=[-4,4]^{n}$.

• Suppose $|Q| \le 1$ and $Q \cap Q_{0} \neq \emptyset$. Then, $Q \subset 5Q_{0}$ and since $p(\cdot )$ satisfies the local log-Hölder continuity condition, $|Q|^{p_{-}(Q)-p_{+}(Q)}\lesssim 1$. Meanwhile, due to Lemma 2.5, since $\sigma (Q_{0}) \lesssim 1$, it follows that $\|\chi _{Q_{0}}\|_{L^{p'(\cdot )}(\sigma )} \lesssim 1$. Thus,

  $$\begin{aligned} \vert Q \vert \lesssim \Vert \chi _{Q} \Vert _{L^{p(\cdot )}(w)} \Vert \chi _{Q} \Vert _{L^{p'( \cdot )}(\sigma )} \lesssim \Vert \chi _{Q} \Vert _{L^{p(\cdot )}(w)} \Vert \chi _{5Q_{0}} \Vert _{L^{p'(\cdot )}(\sigma )} \lesssim \Vert \chi _{Q} \Vert _{L^{p(\cdot )}(w)}. \end{aligned}$$

• Suppose $Q \cap Q_{0} = \emptyset$. Write $x_{Q}$ for the left-lower corner of Q. Let $U_{k}=[-2^{-k}/3,2^{1-k}/3)^{n}$ as before and write $k_{Q}=\max \{k \in {\mathbb{Z}}: U_{k} \cap Q \ne \emptyset \}$. Note that $k_{Q} \le -3$. Then, since $Q \subset U_{k_{Q}-1}$ by Lemma 2.1, we have

  $$\begin{aligned} \vert Q \vert \lesssim \Vert \chi _{Q} \Vert _{L^{p(\cdot )}(w)} \Vert \chi _{Q} \Vert _{L^{p'( \cdot )}(\sigma )} &\lesssim \Vert \chi _{Q} \Vert _{L^{p(\cdot )}(w)} \Vert \chi _{U_{k_{Q}-1}} \Vert _{L^{p'(\cdot )}(\sigma )}\\ &\lesssim \vert U_{k_{Q}-1} \vert \frac{ \Vert \chi _{Q} \Vert _{L^{p(\cdot )}(w)}}{ \Vert \chi _{U_{k_{Q}-1}} \Vert _{L^{p(\cdot )}(w)}}. \end{aligned}$$

  We observe that

  $$\begin{aligned} p_{+}(Q)-p_{-}(Q) = \sup_{y,z \in Q} \bigl\vert p(y)-p(z) \bigr\vert \le 2\sup_{y \in Q} \bigl\vert p(y)-p_{\infty } \bigr\vert \lesssim \frac{1}{\log (e+ \vert x_{Q} \vert )}. \end{aligned}$$

  Thus, since $|U_{k_{Q}-1}| \sim |x_{Q}|^{n}$, we have

  $$ \vert U_{k_{Q}-1} \vert ^{p_{+}(Q)-p_{-}(Q)} \sim \vert x_{Q} \vert ^{\frac{Cn}{\log (e+ \vert x_{Q} \vert )}} = \exp \biggl( \frac{C n\log \vert x_{Q} \vert }{\log (e+ \vert x_{Q} \vert )} \biggr) \lesssim 1 $$

  for some $C>1$. Moreover, since $\|\chi _{U_{k_{Q}-1}}\|_{L^{p(\cdot )}(w)} \ge \|\chi _{U_{0}}\|_{L^{p( \cdot )}(w)} \gtrsim 1$, we obtain the desired result.

• Suppose $|Q| \ge 1$ and $Q \cap Q_{0} \ne \emptyset$. Let $R \in {\mathfrak{D}}$ with $[-1,1]^{n} \subset R$ and $Q \subset R$. Then $\|\chi _{[-1,1]^{n}}\|_{L^{p(\cdot )}(w)} \le \|\chi _{R}\|_{L^{p( \cdot )}(w)} \lesssim \|\chi _{Q}\|_{L^{p(\cdot )}(w)} \le 1$, thanks to Lemma 2.12. Therefore,

  $$ \Vert \chi _{Q} \Vert _{L^{p(\cdot )}(w)}^{p_{-}(Q)-p_{+}(Q)} \lesssim \Vert \chi _{R} \Vert _{L^{p(\cdot )}(w)}^{p_{-}(Q)-p_{+}(Q)} \sim 1. $$

 □

If we fix a cube $P \in {\mathfrak{D}}$ such that $0 \in P$ and if Q is any one of the adjacent cubes to P and has the same size, then $w(Q), \sigma (Q) \gtrsim 1$.

Corollary 2.14

Let $w \in A_{p(\cdot )}(\mathfrak{D})$. Then

$$ \int _{{\mathbb{R}}^{n}}\frac{w(x)\,\mathrm{d}x}{(e+ \vert x \vert )^{K}}, \int _{{\mathbb{R}}^{n}} \frac{\sigma (x)\,\mathrm{d}x}{(e+ \vert x \vert )^{K}}< \infty $$

as long as $K \gg 1$.

Proof

Let $P_{j}$ be the jth parent of P. Then

$$\begin{aligned} \int _{{\mathbb{R}}^{n}}\frac{w(x)\,\mathrm{d}x}{(e+ \vert x \vert )^{K}} = \int _{P_{1}} \frac{w(x)\,\mathrm{d}x}{(e+ \vert x \vert )^{K}} + \sum _{j=1}^{\infty } \int _{P_{j+1} \setminus P_{j}}\frac{w(x)\,\mathrm{d}x}{(e+ \vert x \vert )^{K}} \lesssim \sum _{j=1}^{\infty }2^{-j K}w(P_{j}). \end{aligned}$$

By Lemmas 2.4 and 2.12,

$$\begin{aligned} w(P_{j}) \lesssim \max \bigl( \Vert \chi _{P_{j}} \Vert _{L^{p(\cdot )}(w)}^{p_{-}}, \Vert \chi _{P_{j}} \Vert _{L^{p(\cdot )}(w)}^{p_{+}} \bigr) \lesssim \max \biggl( \biggl( \frac{ \vert P_{j} \vert }{ \vert P \vert } \biggr)^{p_{-}}, \biggl( \frac{ \vert P_{j} \vert }{ \vert P \vert } \biggr)^{p_{+}} \biggr) \lesssim 2^{jnp_{+}}. \end{aligned}$$

Thus, since K is sufficiently large, it follows that

$$\begin{aligned} \int _{{\mathbb{R}}^{n}}\frac{w(x)\,\mathrm{d}x}{(e+ \vert x \vert )^{K}} \lesssim \sum _{j=1}^{\infty }2^{-j(K-np_{+})} \sim 1. \end{aligned}$$

The second inequality is proven similarly. □

Lemma 2.15

Suppose $w \in A_{p(\cdot )}({\mathfrak{D}})$. Let $Q \in {\mathfrak{D}}$ and E be a measurable subset of Q.

• (1) If $w(Q) \ge 1$, then $\|\chi _{Q}\|_{L^{p(\cdot )}(w)} \sim w(Q)^{\frac{1}{p_{\infty }}}$.

• (2) If $w(E) \ge 1$, then

  $$\frac{|E|}{|Q|}\lesssim \biggl(\frac{w(E)}{w(Q)} \biggr)^{\frac{1}{p_{\infty }}} . $$

• (3) In general

  $$\frac{|E|}{|Q|}\lesssim \biggl(\frac{w(E)}{w(Q)} \biggr)^{\frac{1}{p_{+}}}. $$

Proof

• (1) Using Lemma 2.7 for the measure $w(x) \,\mathrm{d}x$, we have

  $$\begin{aligned} \int _{Q} \biggl(\frac{1}{w(Q)^{\frac{1}{p_{\infty }}}} \biggr)^{p(x)}w(x)\,\mathrm{d}x \lesssim \int _{Q} \frac{w(x)\,\mathrm{d}x}{w(Q)} + \int _{Q} \frac{w(x)\,\mathrm{d}x}{(e+ \vert x \vert )^{K}} \lesssim 1. \end{aligned}$$

  Thus, $w(Q) \gtrsim (\|\chi _{Q}\|_{L^{p(\cdot )}(w)})^{p_{\infty }}$. Likewise, due to Lemma 2.7,

  $$\begin{aligned} \int _{Q} \biggl(\frac{1}{ \Vert \chi _{Q} \Vert _{L^{p(\cdot )}(w)}} \biggr)^{p_{\infty }}w(x)\,\mathrm{d}x \lesssim \int _{Q} \frac{w(x)\,\mathrm{d}x}{(\Vert \chi _{Q} \Vert _{L^{p(\cdot )}(w)})^{p(x)}} + \int _{Q} \frac{w(x)\,\mathrm{d}x}{(e+ \vert x \vert )^{K}} \lesssim 1. \end{aligned}$$

  Thus, $w(Q) \lesssim (\|\chi _{Q}\|_{L^{p(\cdot )}(w)})^{p_{\infty }}$.

• (2) As in (1), using Lemma 2.7 for the measure $w(x) \,\mathrm{d}x$, we have

  $$\begin{aligned} \int _{E} \biggl(\frac{1}{w(E)^{\frac{1}{p_{\infty }}}} \biggr)^{p(x)}w(x)\,\mathrm{d}x \lesssim \int _{E} \frac{w(x)\,\mathrm{d}x}{w(E)} + \int _{E} \frac{w(x)\,\mathrm{d}x}{(e+ \vert x \vert )^{K}} \lesssim 1. \end{aligned}$$

  Thus, since $w(Q)^{\frac{1}{p_{\infty }}} \sim \|\chi _{Q}\|_{L^{p(\cdot )}(w)}$,

  $$\begin{aligned} \frac{ \vert E \vert }{ \vert Q \vert } \lesssim \frac{ \Vert \chi _{E} \Vert _{L^{p(\cdot )}(w)}}{ \Vert \chi _{Q} \Vert _{L^{p(\cdot )}(w)}} \lesssim \frac{w(E)^{\frac{1}{p_{\infty }}}}{w(Q)^{\frac{1}{p_{\infty }}}}, \end{aligned}$$

  as required.

• (3) If $w(E) \ge 1$, then this is clear from (2). Suppose $w(E) \le 1$.

  • If $w(Q) \le 1$, then by virtue of Lemma 2.5, $\|\chi _{Q}\|_{L^{p(\cdot )}(w)} \le 1$ and $\|\chi _{E}\|_{L^{p(\cdot )}(w)} \le 1$. Hence

    $$ \bigl( \Vert \chi _{Q} \Vert _{L^{p(\cdot )}(w)}\bigr)^{p_{-}(Q)} \ge w(Q), \qquad \bigl( \Vert \chi _{E} \Vert _{L^{p(\cdot )}(w)}\bigr)^{p_{+}(Q)} \le w(E), $$

    (2.8)

    thanks to Remark 2.6. Meanwhile, $\|\chi _{Q}\|_{L^{p(\cdot )}(w)}{}^{\frac{p_{-}(Q)}{p_{+}(Q)}-1} \lesssim 1$ due to Lemma 2.13. Thus, using (2.8), we have

    $$\begin{aligned} \frac{ \vert E \vert }{ \vert Q \vert } \lesssim \frac{ \Vert \chi _{E} \Vert _{L^{p(\cdot )}(w)}}{ \Vert \chi _{Q} \Vert _{L^{p(\cdot )}(w)}} \lesssim \frac{w(E)^{\frac{1}{p_{+}(Q)}}}{w(Q)^{\frac{1}{p_{+}(Q)}} \Vert \chi _{Q} \Vert _{L^{p(\cdot )}(w)}{}^{1-\frac{p_{-}(Q)}{p_{+}(Q)}}} \lesssim \frac{w(E)^{\frac{1}{p_{+}}}}{w(Q)^{\frac{1}{p_{+}}}}. \end{aligned}$$

  • If $w(E) \le 1 \le w(Q)$, then

    $$ \bigl( \Vert \chi _{Q} \Vert _{L^{p(\cdot )}(w)} \bigr)^{p_{+}(Q)} \ge w(Q), \qquad \bigl( \Vert \chi _{E} \Vert _{L^{p(\cdot )}(w)}\bigr)^{p_{+}(Q)} \le w(E), $$

    thanks to Remark 2.6. Thus,

    $$\begin{aligned} \frac{ \vert E \vert }{ \vert Q \vert } \lesssim \frac{ \Vert \chi _{E} \Vert _{L^{p(\cdot )}(w)}}{ \Vert \chi _{Q} \Vert _{L^{p(\cdot )}(w)}} \le \frac{w(E)^{\frac{1}{p_{+}(Q)}}}{w(Q)^{\frac{1}{p_{+}(Q)}}} \lesssim \frac{w(E)^{\frac{1}{p_{+}}}}{w(Q)^{\frac{1}{p_{+}}}}. \end{aligned}$$

 □

Next, we consider some estimates related to the dyadic maximal operator. We define

$$ m_{Q}(g)=\frac{1}{ \vert Q \vert } \int _{Q} g(y) {\,\mathrm{d}}y \quad \bigl(g \in L^{1}_{ \mathrm{loc}}\bigl({\mathbb{R}}^{n}\bigr)\bigr) $$

for a cube Q.

Lemma 2.16

(Sparse decomposition of f, [15, p. 250])

Let $f \in L^{\infty }_{\mathrm{c}}({\mathbb{R}}^{n}) \setminus \{0\}$, $\lambda >0$, $a \gg 2^{n}$, and let ${\mathfrak{D}}$ be a dyadic grid. Then there exists a set of pairwise disjoint dyadic cubes $\{Q_{j}^{k}\}_{k\in \mathbb{Z}, j \in J_{k}}$ such that

$$ \Omega _{k} \equiv \bigl\{ x \in {\mathbb{R}}^{n} : M_{{\mathfrak{D}}}f(x)>a^{k} \bigr\} = \bigcup _{j \in J_{k}}Q_{j}^{k}. $$

Further, these cubes have the property of $a^{k}< m_{Q_{j}^{k}}(|f|)\le 2^{n} a^{k}$ for all $j \in J_{k}$. Furthermore, there exists a disjoint collection $\{E_{j}^{k}\}_{k \in {\mathbb{Z}}, j \in J_{k}}$, where each $E_{j}^{k}$ is a subset of $Q_{j}^{k}$ called a nutshell, such that $2|E_{j}^{k}| \ge |Q_{j}^{k}|$ and that

$$ \Omega _{k} \setminus \Omega _{k+1}=\bigcup _{j \in J_{k}}E_{j}^{k}. $$

A direct consequence of Lemma 2.16 is that

$$ M_{{\mathfrak{D}}}f \lesssim \sum_{k=-\infty }^{\infty } \sum_{j \in J_{k}}m_{Q_{j}^{k}}\bigl( \vert f \vert \bigr) \chi _{E_{j}^{k}}. $$

Given a weight W and a measurable function f, define

$$ M_{W,{\mathfrak{D}}}f(x)= \sup_{Q \in {\mathfrak{D}}} \frac{\chi _{Q}(x)}{W(Q)} \int _{Q} \bigl\vert f(y) \bigr\vert W(y)\,\mathrm{d}y. $$

The next lemma reflects the geometric property of ${\mathfrak{D}}$.

Lemma 2.17

([10, Lemma 2.1])

Given a weight W and $1< p<\infty$, we have

$$ \int _{{\mathbb{R}}^{n}}M_{W,{\mathfrak{D}}}f(x)^{p}W(x)\,\mathrm{d}x \le \frac{p \cdot 2^{p}}{p-1} \int _{{\mathbb{R}}^{n}} \bigl\vert f(x) \bigr\vert ^{p}W(x)\,\mathrm{d}x. $$

We transform Lemma 2.17 as shown below.

Lemma 2.18

Let $\{Q_{j}^{k}\}_{k \in {\mathbb{Z}}, j \in J_{k}}$ be a sparse collection with the nutshell $\{E_{j}^{k}\}_{k \in {\mathbb{Z}}, j \in J_{k}}$, and let $1< r<\infty$. Also let $W \in A_{\infty }({\mathfrak{D}})$. Then for all nonnegative $f \in L^{0}({\mathbb{R}}^{n})$,

$$ \sum_{k=-\infty }^{\infty }\sum _{j \in J_{k}} \biggl( \frac{1}{W(Q_{j}^{k})} \int _{Q_{j}^{k}}f(y)W(y)\,\mathrm{d}y \biggr)^{r} W \bigl(Q_{j}^{k}\bigr) \lesssim \int _{{\mathbb{R}}^{n}}f(x)^{r}W(x)\,\mathrm{d}x. $$

Proof

By Lemmas 2.9 and 2.17,

$$\begin{aligned} &\sum_{k=-\infty }^{\infty }\sum _{j \in J_{k}} \biggl( \frac{1}{W(Q_{j}^{k})} \int _{Q_{j}^{k}}f(y)W(y)\,\mathrm{d}y \biggr)^{r} W \bigl(Q_{j}^{k}\bigr) \\ &\quad \lesssim \sum_{k=-\infty }^{\infty }\sum _{j \in J_{k}} \biggl( \frac{1}{W(Q_{j}^{k})} \int _{Q_{j}^{k}}f(y)W(y)\,\mathrm{d}y \biggr)^{r} W \bigl(E_{j}^{k}\bigr) \\ &\quad = \sum_{k=-\infty }^{\infty }\sum _{j \in J_{k}} \int _{E_{j}^{k}} \biggl( \frac{1}{W(Q_{j}^{k})} \int _{Q_{j}^{k}}f(y)W(y)\,\mathrm{d}y \biggr)^{r} W(x)\,\mathrm{d}x \\ &\quad \le \sum_{k=-\infty }^{\infty }\sum _{j \in J_{k}} \int _{E_{j}^{k}} M_{W,{ \mathcal{D}}}f(x)^{r} W(x)\,\mathrm{d}x \\ &\quad \lesssim \int _{{\mathbb{R}}^{n}}f(x)^{r} W(x)\,\mathrm{d}x, \end{aligned}$$

as required. □

Lemma 2.19

Let $w \in A_{p(\cdot )}({\mathfrak{D}})$ and $Q \in {\mathfrak{D}}$.

• (1) $\|\chi _{Q}\|_{L^{p'(\cdot )}(\sigma )}{}^{p_{-}(Q)-p(x)} \lesssim 1$ for all $x \in Q$.

• (2) We have

  $$\biggl( \frac{\sigma (Q)}{\|\chi _{Q}\|_{L^{p'(\cdot )}(\sigma )}} \biggr)^{p_{-}(Q)} \le \sigma (Q) . $$

• (3) $\int _{Q}\sigma (Q)^{p_{-}(Q)}|Q|^{-p(x)}w(x)\,\mathrm{d}x \lesssim \sigma (Q)$.

Proof

Note that $p_{+}'=(p')_{-}$ and $p_{-}'=(p')_{+}$, where $(p')_{+}$ and $(p')_{-}$ are the supremum and the infimum of $p'(\cdot )$, respectively.

• (1) Since $p(x)-p_{-}(Q) \lesssim (p')_{+}(Q)-(p')_{-}(Q)$ as in [3, p. 755], we can use Lemma 2.13.

• (2) If $\|\chi _{Q}\|_{L^{p'(\cdot )}(\sigma )} \ge 1$, then $\|\chi _{Q}\|_{L^{p'(\cdot )}(\sigma )} \ge \sigma (Q)^{ \frac{1}{(p')_{+}(Q)}}$ by Remark 2.6. Hence

  $$ \biggl(\frac{\sigma (Q)}{ \Vert \chi _{Q} \Vert _{L^{p'(\cdot )}(\sigma )}} \biggr)^{p_{-}(Q)} \le \biggl( \frac{\sigma (Q)}{\sigma (Q)^{\frac{1}{(p')_{+}(Q)}}} \biggr)^{p_{-}(Q)} = \biggl(\frac{\sigma (Q)}{\sigma (Q)^{\frac{1}{p_{-}'(Q)}}} \biggr)^{p_{-}(Q)} = \sigma (Q). $$

  If $\|\chi _{Q}\|_{L^{p'(\cdot )}(\sigma )} \le 1$, then, again by Remark 2.6,

  $$ \sigma (Q)^{\frac{1}{(p')_{+}(Q)}} \ge \Vert \chi _{Q} \Vert _{L^{p'(\cdot )}( \sigma )} \ge \sigma (Q)^{\frac{1}{(p')_{-}(Q)}}. $$

  Therefore

  $$\begin{aligned} \biggl( \frac{\sigma (Q)}{ \Vert \chi _{Q} \Vert _{L^{p'(\cdot )}(\sigma )}} \biggr)^{p_{-}(Q)} &\le \bigl( \Vert \chi _{Q} \Vert _{L^{p'(\cdot )}(\sigma )}{}^{(p')_{-}(Q)-1} \bigr)^{p_{-}(Q)} \\ &\le \bigl( \Vert \chi _{Q} \Vert _{L^{p'(\cdot )}(\sigma )}{}^{(p')_{+}(Q)-1} \bigr)^{p_{-}(Q)} \\ &\le \bigl(\sigma (Q)^{\frac{(p')_{+}(Q)-1}{(p')_{+}(Q)}} \bigr)^{p_{-}(Q)} \\ &= \bigl(\sigma (Q)^{\frac{p'_{-}(Q)-1}{p'_{-}(Q)}} \bigr)^{p_{-}(Q)} \\ &=\sigma (Q), \end{aligned}$$

  as required.

• (3) By the definition of $A_{p(\cdot )}({\mathfrak{D}})$, we have

  $$ \bigl\Vert \bigl([w]_{A_{p(\cdot )}({\mathfrak{D}})} \vert Q \vert \bigr)^{-1} \bigr\Vert \chi _{Q}\|_{L^{p'(\cdot )}(\sigma )}\chi _{Q}\| _{L^{p(\cdot )}(w)} \le 1, $$

  or equivalently,

  $$ \int _{Q} \biggl( \frac{ \Vert \chi _{Q} \Vert _{L^{p'(\cdot )}(\sigma )}}{[w]_{A_{p(\cdot )}({\mathfrak{D}})} \vert Q \vert } \biggr)^{p(x)} w(x)\,\mathrm{d}x \le 1 $$

  (2.9)

  for $Q \in {\mathfrak{D}}$. From (1) and (2) we deduce

  $$\begin{aligned} & \int _{Q}\sigma (Q)^{p_{-}(Q)} \vert Q \vert ^{-p(x)}w(x)\,\mathrm{d}x \\ &\quad = \biggl(\frac{\sigma (Q)}{ \Vert \chi _{Q} \Vert _{L^{p'(\cdot )}(\sigma )}} \biggr)^{p_{-}(Q)} \int _{Q} \Vert \chi _{Q} \Vert _{L^{p'(\cdot )}(\sigma )}{}^{p_{-}(Q)-p(x)} \Vert \chi _{Q} \Vert _{L^{p'(\cdot )}(\sigma )}{}^{p(x)} \vert Q \vert ^{-p(x)}w(x)\,\mathrm{d}x \\ &\quad \lesssim \sigma (Q) \int _{Q} \Vert \chi _{Q} \Vert _{L^{p'(\cdot )}(\sigma )}{}^{p(x)} \vert Q \vert ^{-p(x)}w(x)\,\mathrm{d}x \lesssim \sigma (Q). \end{aligned}$$

 □

Lemma 2.20

Suppose that $w \in A_{p(\cdot )}({\mathfrak{D}})$. Let ${\mathcal{S}}$ be a disjoint collection of cubes in dyadic grid ${\mathfrak{D}}$. Then

$$\begin{aligned} \sum_{Q \in {\mathcal{S}}} \int _{Q}\bigl(e+ \vert x \vert \bigr)^{-K} \sigma (Q)^{p_{-}(Q)} \vert Q \vert ^{-p(x)}w(x)\,\mathrm{d}x \lesssim 1. \end{aligned}$$

Proof

Let $Q \in {\mathcal{S}}$. Then either $Q \supset P_{l}$ for some l or Q is included in some $P_{l}$. Since the first possibility can occur only in one cube, we only have to consider the second possibility. For such a cube Q, we let l be the smallest number such that $Q \subset P_{l}$. Then $P_{l-1}$ and Q never intersect due to the minimality of l, since Q does not contain $P_{l}$. Thus, there exists uniquely an integer l such that $Q \subset P_{l} \setminus P_{l-1}$.

Using Lemma 2.19(3) and Corollary 2.14, we estimate

$$\begin{aligned} &\sum_{Q \in {\mathcal{S}}} \int _{Q}\bigl(e+ \vert x \vert \bigr)^{-K} \sigma (Q)^{p_{-}(Q)} \vert Q \vert ^{-p(x)}w(x)\,\mathrm{d}x \\ &\quad = \sum_{l=1}^{\infty }\sum _{Q \in {\mathcal{S}}, Q \subset P_{l} \setminus P_{l-1}} \int _{Q}\bigl(e+ \vert x \vert \bigr)^{-K} \sigma (Q)^{p_{-}(Q)} \vert Q \vert ^{-p(x)}w(x)\,\mathrm{d}x \\ &\quad \lesssim \sum_{l=1}^{\infty }2^{-K l} \sum_{Q \in {\mathcal{S}}, Q \subset P_{l} \setminus P_{l-1}} \int _{Q} \sigma (Q)^{p_{-}(Q)} \vert Q \vert ^{-p(x)}w(x)\,\mathrm{d}x \\ &\quad \lesssim \sum_{l=1}^{\infty }2^{-K l} \sum_{Q \in {\mathcal{S}}, Q \subset P_{l} \setminus P_{l-1}} \sigma (Q) \\ &\quad \lesssim \sum_{l=1}^{\infty }\sum _{Q \in {\mathcal{S}}, Q \subset P_{l} \setminus P_{l-1}} \int _{Q}\frac{\sigma (x)\,\mathrm{d}x}{(e+ \vert x \vert )^{K}} \\ &\quad \lesssim 1. \end{aligned}$$

 □

The next lemma is used in Sect. 3.2.

Lemma 2.21

Suppose that $w \in A_{p(\cdot )}({\mathfrak{D}})$. Let $\{Q_{j}^{k}\}_{k \in {\mathbb{Z}}, j \in J_{k}}$ be a sparse collection with the nutshell $\{E_{j}^{k}\}_{k \in {\mathbb{Z}}, j \in J_{k}}$. Set ${\mathcal{H}}_{2} \equiv \{ (k,j) : Q_{j}^{k} \cap P= \phi , \sigma (Q_{j}^{k}) \ge 1 \}$. Then

$$ \sum_{(k,j) \in {\mathcal{H}}_{2}} \int _{E_{j}^{k}} \biggl( \frac{ \Vert \chi _{Q_{j}^{k}} \Vert _{L^{p'(\cdot )}(\sigma )}}{ \vert Q_{j}^{k} \vert } \biggr)^{p(x)}\frac{w(x)\,\mathrm{d}x}{(e+ \vert x \vert )^{K}} \lesssim 1. $$

Proof

By (2.7), we obtain

$$\begin{aligned} &\sum_{(k,j) \in {\mathcal{H}}_{2}} \int _{E_{j}^{k}} \biggl( \frac{ \Vert \chi _{Q_{j}^{k}} \Vert _{L^{p'(\cdot )}(\sigma )}}{ \vert Q_{j}^{k} \vert } \biggr)^{p(x)}\frac{w(x)\,\mathrm{d}x}{(e+ \vert x \vert )^{K}} \\ &\quad \le \sum_{(k,j) \in {\mathcal{H}}_{2}} \sup_{x \in Q_{j}^{k}} \frac{1}{(e+ \vert x \vert )^{K}} \int _{Q_{j}^{k}} \biggl( \frac{ \Vert \chi _{Q_{j}^{k}} \Vert _{L^{p'(\cdot )}(\sigma )}}{ \vert Q_{j}^{k} \vert } \biggr)^{p(x)}w(x)\,\mathrm{d}x \quad \lesssim \sum_{(k,j) \in {\mathcal{H}}_{2}} \sup_{x \in Q_{j}^{k}} \frac{1}{(e+ \vert x \vert )^{K}}. \end{aligned}$$

By Lemma 2.9, Corollary 2.14, and the fact that $\sigma (Q_{j}^{k})\ge 1$ for any $(k,j) \in {\mathcal{H}}_{2}$, we obtain

$$\begin{aligned} &\sum_{(k,j) \in {\mathcal{H}}_{2}} \int _{E_{j}^{k}} \biggl( \frac{ \Vert \chi _{Q_{j}^{k}} \Vert _{L^{p'(\cdot )}(\sigma )}}{ \vert Q_{j}^{k} \vert } \biggr)^{p(x)}\frac{w(x)\,\mathrm{d}x}{(e+ \vert x \vert )^{K}} \\ &\quad \lesssim \sum _{(k,j) \in { \mathcal{H}}_{2}} \sup_{x \in Q_{j}^{k}}\frac{1}{(e+ \vert x \vert )^{K}} \sigma \bigl(Q_{j}^{k}\bigr) \\ &\quad \lesssim \sum_{(k,j) \in {\mathcal{H}}_{2}} \int _{Q_{j}^{k}} \frac{\sigma (x)}{(e+ \vert x \vert )^{K}} {\,\mathrm{d}}x \\ &\quad \lesssim \sum_{(k,j) \in {\mathcal{H}}_{2}} \int _{E_{j}^{k}} \frac{\sigma (x)}{(e+ \vert x \vert )^{K}} {\,\mathrm{d}}x \\ &\quad \lesssim 1. \end{aligned}$$

 □

3 Proof of Theorem 1.7

Sect. 3 uses the notation in Lemma 2.21.

Suppose that $w \in A_{p(\cdot )}({\mathfrak{D}})$ and $f \in L^{\infty }_{\mathrm{c}}({\mathbb{R}}^{n})$ satisfy $\|f\|_{L^{p(\cdot )}(w)}<1$. We may assume $f\ge 0$ a.e. by replacing f by $|f|$ if necessary. We write $f_{1}\equiv f\chi _{\{f>\sigma \}}$ and $f_{2}\equiv f-f_{1}$. Here, $\sigma \equiv w^{-\frac{1}{p(\cdot )-1}}$ is the dual weight. Then

$$ \int _{{\mathbb{R}}^{n}} \bigl\vert f_{j}(x) \bigr\vert ^{p(x)}w(x)\,\mathrm{d}x< 1 \quad (j=1,2). $$

(3.1)

We only have to show that for $j=1,2$,

$$ \int _{{\mathbb{R}}^{n}}M_{\mathfrak{D}} f_{j}(x)^{p(x)}w(x)\,\mathrm{d}x \lesssim 1. $$

We form a sparse decomposition of $f_{1}$ and $f_{2}$ separately. The estimates of $f_{1}$ and $f_{2}$ will be established independently. So suppose that there exists a sparse family $\{Q_{j}^{k}\}_{k \in {\mathbb{Z}}, j \in J_{k}}$ with the nutshell $\{E_{j}^{k}\}_{k \in {\mathbb{Z}}, j \in J_{k}}$ such that

$$ M_{\mathfrak{D}} f_{l}\lesssim \sum _{k=-\infty }^{\infty }\sum_{j \in J_{k}}m_{Q_{j}^{k}} \bigl( \vert f_{l} \vert \bigr) \chi _{E_{j}^{k}} \quad (l=1,2). $$

Since the $E_{j}^{k}$’s are disjoint, we have

$$\begin{aligned} \int _{{\mathbb{R}}^{n}} \bigl( M_{\mathfrak{D}} f_{l}(x) \bigr)^{p(x)}w(x) \,\mathrm{d}x \lesssim \sum_{k=-\infty }^{\infty } \sum_{j \in J_{k}} \int _{E_{j}^{k}}m_{Q_{j}^{k}}\bigl( \vert f_{l} \vert \bigr)^{p(x)} w(x)\,\mathrm{d}x =:{ \mathrm{I}}_{l} \quad (l=1,2). \end{aligned}$$

3.1 Estimate of $M_{\mathfrak{D}} f_{1}$

We use the sparse decomposition of $M_{\mathfrak{D}} f_{1}$:

$$ M_{\mathfrak{D}} f_{1}\lesssim \sum _{k=-\infty }^{\infty }\sum_{j \in J_{k}} m_{Q_{j}^{k}}\bigl( \vert f_{1} \vert \bigr)\chi _{E_{j}^{k}}. $$

Let $x \in {\mathbb{R}}^{n}$. Note that $f_{1}(x)\sigma ^{-1}(x) \ge 1$ unless $f_{1}(x)=0$. Since $p_{-}(Q_{j}^{k}) \ge 1$,

$$\begin{aligned} \int _{Q_{j}^{k}}\bigl(f_{1}(y)\sigma (y)^{-1}\bigr)^{ \frac{p(y)}{p_{-}(Q_{j}^{k})}}\sigma (y)\,\mathrm{d}y &\le \int _{Q_{j}^{k}}\bigl(f_{1}(y) \sigma (y)^{-1}\bigr)^{p(y)}\sigma (y)\,\mathrm{d}y \\ &= \int _{Q_{j}^{k}}f_{1}(y)^{p(y)}w(y)\,\mathrm{d}y \le 1. \end{aligned}$$

Consequently,

$$\begin{aligned} \biggl( \int _{Q_{j}^{k}}\bigl(f_{1}(y)\sigma (y)^{-1}\bigr)\sigma (y)\,\mathrm{d}y \biggr)^{p(x)} &\le \biggl( \int _{Q_{j}^{k}}\bigl(f_{1}(y)\sigma (y)^{-1}\bigr)^{ \frac{p(y)}{p_{-}(Q_{j}^{k})}}\sigma (y)\,\mathrm{d}y \biggr)^{p(x)} \\ &\le \biggl( \int _{Q_{j}^{k}}\bigl(f_{1}(y)\sigma (y)^{-1}\bigr)^{ \frac{p(y)}{p_{-}(Q_{j}^{k})}}\sigma (y)\,\mathrm{d}y \biggr)^{p_{-}(Q_{j}^{k})}. \end{aligned}$$

Hence, by Lemma 2.19(3),

$$\begin{aligned} {\mathrm{I}}_{1} &\le \sum_{k=-\infty }^{\infty } \sum_{j \in J_{k}} \biggl( \frac{1}{\sigma (Q_{j}^{k})} \int _{Q_{j}^{k}}\bigl(f_{1}(y)\sigma (y)^{-1}\bigr)^{ \frac{p(y)}{p_{-}(Q_{j}^{k})}}\sigma (y)\,\mathrm{d}y \biggr)^{p_{-}(Q_{j}^{k})} \\ & \quad \times \int _{E_{j}^{k}} \sigma \bigl(Q_{j}^{k} \bigr)^{p_{-}(Q_{j}^{k})} \bigl\vert Q_{j}^{k} \bigr\vert ^{-p(x)}w(x)\,\mathrm{d}x \\ &\lesssim \sum_{k=-\infty }^{\infty }\sum _{j \in J_{k}} \biggl( \frac{1}{\sigma (Q_{j}^{k})} \int _{Q_{j}^{k}}\bigl(f_{1}(y)\sigma (y)^{-1}\bigr)^{ \frac{p(y)}{p_{-}(Q_{j}^{k})}}\sigma (y)\,\mathrm{d}y \biggr)^{p_{-}(Q_{j}^{k})} \sigma \bigl(Q_{j}^{k}\bigr) \\ &\le \sum_{k=-\infty }^{\infty }\sum _{j \in J_{k}} \biggl( \frac{1}{\sigma (Q_{j}^{k})} \int _{Q_{j}^{k}}\bigl(f_{1}(y)\sigma (y)^{-1}\bigr)^{ \frac{p(y)}{p_{-}}}\sigma (y)\,\mathrm{d}y \biggr)^{p_{-}} \sigma \bigl(Q_{j}^{k}\bigr). \end{aligned}$$

The last inequality used Hölder’s inequality. Since $\sigma \in A_{\infty }({\mathfrak{D}})$, $\sigma (Q_{j}^{k}) \lesssim \sigma (E_{j}^{k})$ by virtue of Lemma 2.9. Consequently, due to Lemma 2.18, we have

$$\begin{aligned} I_{1} \lesssim \int _{{\mathbb{R}}^{n}} \bigl\{ \bigl[f_{1}(x) \sigma ^{-1}(x)\bigr]^{ \frac{p(x)}{p_{-}}} \bigr\} ^{p_{-}}\sigma (x)\,\mathrm{d}x \le 1. \end{aligned}$$

3.2 Estimate of $M f_{2}$

Set

$$\begin{aligned}& {\mathcal{F}} \equiv \bigl\{ (k,j):Q_{j}^{k} \subset P\bigr\} , \\& { \mathcal{G}} \equiv \bigl\{ (k,j):P \subset Q_{j}^{k}\bigr\} , \\& { \mathcal{H}} \equiv \bigl\{ (k,j):P \cap Q_{j}^{k} = \emptyset \bigr\} . \end{aligned}$$

Accordingly,

$$\begin{aligned} {\mathrm{I}}_{2,{\mathcal{A}}} \equiv \sum_{(k,j) \in {\mathcal{A}}} \int _{E_{j}^{k}}m_{Q_{j}^{k}}\bigl( \vert f_{2} \vert \bigr)^{p(x)} w(x)\,\mathrm{d}x \quad \bigl(\mathcal{A} \in \{\mathcal{F}, \mathcal{G}, \mathcal{H} \}\bigr). \end{aligned}$$

Estimate of ${\mathrm{I}}_{2,{\mathcal{F}}}$

Since $f_{2} \sigma ^{-1} \le 1$, we have

$$\begin{aligned} {\mathrm{I}}_{2,{\mathcal{F}}} &= \sum_{(k,j) \in {\mathcal{F}}} \int _{E_{j}^{k}} \biggl( \frac{1}{ \vert Q_{j}^{k} \vert } \int _{Q_{j}^{k}} f_{2}(y)\sigma (y)^{-1} \sigma (y) {\,\mathrm{d}}y \biggr)^{p(x)} w(x) {\,\mathrm{d}}x \\ &\le \sum_{(k,j) \in {\mathcal{F}}} \int _{E_{j}^{k}} \sigma \bigl(Q_{j}^{k} \bigr)^{p(x)-p_{-}(Q_{j}^{k})} \sigma \bigl(Q_{j}^{k} \bigr)^{p_{-}(Q_{j}^{k})} \bigl\vert Q_{j}^{k} \bigr\vert ^{-p(x)}w(x)\,\mathrm{d}x \\ &\le \sum_{(k,j) \in {\mathcal{F}}} \int _{E_{j}^{k}} \bigl(1+\sigma \bigl(Q_{j}^{k} \bigr)\bigr)^{p(x)-p_{-}(Q_{j}^{k})} \sigma \bigl(Q_{j}^{k} \bigr)^{p_{-}(Q_{j}^{k})} \bigl\vert Q_{j}^{k} \bigr\vert ^{-p(x)}w(x)\,\mathrm{d}x \\ &\le \bigl(1+\sigma (P)\bigr)^{p_{+}-p_{-}} \sum _{(k,j) \in {\mathcal{F}}} \int _{E_{j}^{k}} \sigma \bigl(Q_{j}^{k} \bigr)^{p_{-}(Q_{j}^{k})} \bigl\vert Q_{j}^{k} \bigr\vert ^{-p(x)}w(x)\,\mathrm{d}x. \end{aligned}$$

From Lemmas 2.19(3) and 2.9, we obtain

$$\begin{aligned} {\mathrm{I}}_{2,{\mathcal{F}}}&\lesssim \bigl(1+\sigma (P)\bigr)^{p_{+}-p_{-}} \sum_{(k,j) \in {\mathcal{F}}} \sigma \bigl(Q_{j}^{k} \bigr) \\ &\lesssim \bigl(1+\sigma (P)\bigr)^{p_{+}-p_{-}} \sum _{(k,j) \in {\mathcal{F}}} \sigma \bigl(E_{j}^{k} \bigr) \lesssim \bigl(1+\sigma (P)\bigr)^{p_{+}-p_{-}}\sigma (P) \lesssim 1. \end{aligned}$$

Estimate of ${\mathrm{I}}_{2,{\mathcal{G}}}$

We note that $w(Q_{j}^{k}) \ge w(P) \ge 1$ and $\sigma (Q_{j}^{k})\ge \sigma (P) \ge 1$. Consequently, from Lemma 2.15(1)–(2),

$$ \frac{1}{ \vert Q_{j}^{k} \vert } \lesssim \frac{1}{ \vert P \vert } \sigma (P)^{ \frac{1}{p_{\infty }'}} \sigma \bigl(Q_{j}^{k}\bigr)^{-\frac{1}{p_{\infty }'}} \lesssim \sigma \bigl(Q_{j}^{k}\bigr)^{-\frac{1}{p_{\infty }'}} \lesssim \frac{1}{ \Vert \chi _{Q_{j}^{k}} \Vert _{L^{p'(\cdot )}(\sigma )}}. $$

Thus, by Hölder’s inequality and (3.1),

$$\begin{aligned} m_{Q_{j}^{k}}(f_{2}) \lesssim \frac{1}{ \Vert \chi _{Q_{j}^{k}} \Vert _{L^{p'(\cdot )}(\sigma )}} \int _{Q_{j}^{k}}f_{2}(y)\,\mathrm{d}y &\lesssim \frac{ \Vert f_{2} \Vert _{L^{p(\cdot )}(w)} \Vert \chi _{Q_{j}^{k}} \Vert _{L^{p'(\cdot )}(\sigma )}}{ \Vert \chi _{Q_{j}^{k}} \Vert _{L^{p'(\cdot )}(\sigma )}} \lesssim 1. \end{aligned}$$

Using Lemma 2.7 for the measure $w(x) \,\mathrm{d}x$, we have

$$\begin{aligned} {\mathrm{I}}_{2,{\mathcal{G}}} &\lesssim \sum_{(k,j) \in {\mathcal{G}}} \int _{E_{j}^{k}} \bigl( m_{Q_{j}^{k}}(f_{2}) \bigr)^{p_{\infty }}w(x)\,\mathrm{d}x + \sum_{(k,j) \in {\mathcal{G}}} \int _{E_{j}^{k}} \frac{w(x)\,\mathrm{d}x}{(e+ \vert x \vert )^{K}} \\ &\lesssim \sum_{(k,j) \in {\mathcal{G}}} \int _{E_{j}^{k}} \bigl( m_{Q_{j}^{k}}(f_{2}) \bigr)^{p_{\infty }}w(x)\,\mathrm{d}x +1. \end{aligned}$$

To complete the estimate of ${\mathrm{I}}_{2,{\mathcal{G}}}$, we only have to show that the first sum is bounded by a constant. We calculate

$$\begin{aligned} &\sum_{(k,j) \in {\mathcal{G}}} \int _{E_{j}^{k}} \bigl(m_{Q_{j}^{k}}(f_{2}) \bigr)^{p_{\infty }}w(x)\,\mathrm{d}x \\ &\quad = \sum_{(k,j) \in {\mathcal{G}}} w\bigl(E_{j}^{k} \bigr) \biggl( \frac{\sigma (Q_{j}^{k})}{ \vert Q_{j}^{k} \vert } \biggr)^{p_{\infty }} \biggl( \frac{1}{\sigma (Q_{j}^{k})} \int _{Q_{j}^{k}}f_{2}(y)\sigma (y)^{-1} \sigma (y)\,\mathrm{d}y \biggr)^{p_{\infty }}. \end{aligned}$$

We note that

$$ \sigma \bigl(Q_{j}^{k}\bigr)^{p_{\infty }-1} = \sigma \bigl(Q_{j}^{k}\bigr)^{ \frac{p_{\infty }}{p_{\infty }'}} \sim \bigl( \Vert \chi _{Q_{j}^{k}} \Vert _{L^{p'( \cdot )}(\sigma )} \bigr)^{p_{\infty }} \lesssim \biggl( \frac{ \vert Q_{j}^{k} \vert }{ \Vert \chi _{Q_{j}^{k}} \Vert _{L^{p(\cdot )}(w)}} \biggr)^{p_{\infty }} \sim \frac{ \vert Q_{j}^{k} \vert ^{p_{\infty }}}{w(Q_{j}^{k})} $$

thanks to Lemma 2.15(1) and the definition of $A_{p(\cdot )}({\mathfrak{D}})$. Thus,

$$\begin{aligned} &\sum_{(k,j) \in {\mathcal{G}}} \int _{E_{j}^{k}} \bigl(m_{Q_{j}^{k}}(f_{2}) \bigr)^{p_{\infty }}w(x)\,\mathrm{d}x \\ &\quad = \sum_{(k,j) \in {\mathcal{G}}} w\bigl(E_{j}^{k} \bigr) \biggl( \frac{\sigma (Q_{j}^{k})}{ \vert Q_{j}^{k} \vert } \biggr)^{p_{\infty }} \biggl( \frac{1}{\sigma (Q_{j}^{k})} \int _{Q_{j}^{k}}f_{2}(y)\sigma (y)^{-1} \sigma (y)\,\mathrm{d}y \biggr)^{p_{\infty }} \\ &\quad \lesssim \sum_{(k,j) \in {\mathcal{G}}} \sigma \bigl(Q_{j}^{k} \bigr) \frac{w(E_{j}^{k})}{w(Q_{j}^{k})} \biggl( \frac{1}{\sigma (Q_{j}^{k})} \int _{Q_{j}^{k}}f_{2}(y)\sigma (y)^{-1} \sigma (y)\,\mathrm{d}y \biggr)^{p_{\infty }} \\ &\quad \lesssim \sum_{(k,j) \in {\mathcal{G}}} \sigma \bigl(E_{j}^{k} \bigr) \biggl( \frac{1}{\sigma (Q_{j}^{k})} \int _{Q_{j}^{k}}f_{2}(y)\sigma (y)^{-1} \sigma (y)\,\mathrm{d}y \biggr)^{p_{\infty }} \\ &\quad \lesssim \int _{{\mathbb{R}}^{n}}\bigl(f_{2}(x)\sigma (x)^{-1}\bigr)^{p_{\infty }} \sigma (x)\,\mathrm{d}x, \end{aligned}$$

owing to Lemma 2.18. Since $0 \le f_{2} \sigma ^{-1} \le 1$, we have

$$\begin{aligned} \int _{{\mathbb{R}}^{n}}\bigl(f_{2}(x)\sigma (x)^{-1}\bigr)^{p_{\infty }}\sigma (x)\,\mathrm{d}x &\lesssim \int _{{\mathbb{R}}^{n}}\bigl(f_{2}(x)\sigma (x)^{-1}\bigr)^{p(x)} \sigma (x)\,\mathrm{d}x + \int _{{\mathbb{R}}^{n}}\frac{\sigma (x)}{(e+ \vert x \vert )^{K}}\,\mathrm{d}x \\ &\lesssim \int _{{\mathbb{R}}^{n}}f_{2}(x)^{p(x)}w(x)\,\mathrm{d}x+1 \lesssim 1, \end{aligned}$$

thanks to Lemma 2.7 applied for the weighted measure $\sigma (x)\,\mathrm{d}x$. Consequently, ${\mathrm{I}}_{2,{\mathcal{G}}} \lesssim 1$.

Estimate of ${\mathrm{I}}_{3,{\mathcal{G}}}$

Set

$$\begin{aligned}& {\mathcal{H}}_{1} \equiv \bigl\{ (k,j) \in {\mathcal{H}}:\sigma \bigl(Q_{j}^{k}\bigr) \le 1\bigr\} , \qquad { \mathcal{H}}_{2} \equiv \bigl\{ (k,j) \in {\mathcal{H}}: \sigma \bigl(Q_{j}^{k}\bigr)>1\bigr\} . \end{aligned}$$

Accordingly, we consider

$$\begin{aligned} {\mathrm{I}}_{2,{\mathcal{H}}_{l}} \equiv \sum_{(k,j) \in {\mathcal{H}}_{l}} \int _{E_{j}^{k}}m_{Q_{j}^{k}}\bigl( \vert f_{2} \vert \bigr)^{p(x)} w(x)\,\mathrm{d}x\quad (l=1,2). \end{aligned}$$

Let $(k,j) \in {\mathcal{H}}_{1}$. Let $x_{+} \in \overline{Q_{j}^{k}}$ satisfy $p(x_{+})=p_{+}(Q_{j}^{k})$. Then

$$\begin{aligned} \bigl\vert p_{+}\bigl(Q_{j}^{k} \bigr)-p(x) \bigr\vert \le \bigl\vert p(x)-p_{\infty } \bigr\vert + \bigl\vert p(x_{+})-p_{\infty } \bigr\vert \lesssim \frac{1}{\log (e+ \vert x \vert )} \quad \bigl(x \in Q_{j}^{k} \bigr). \end{aligned}$$

Also see [3, (5.14)]. Consequently, from Lemma 2.7 applied for the measure $w(x) \,\mathrm{d}x$, we deduce

$$\begin{aligned} {\mathrm{I}}_{2,{\mathcal{H}}_{1}} &\lesssim \sum_{(k,j) \in {\mathcal{H}}_{1}} \int _{E_{j}^{k}} m_{Q_{j}^{k}}(f_{2})^{p_{+}(Q_{j}^{k})}w(x)\,\mathrm{d}x + \sum_{(k,j) \in {\mathcal{H}}_{1}} \int _{E_{j}^{k}} \frac{w(x)\,\mathrm{d}x}{(e+ \vert x \vert )^{K}} \\ &\lesssim \sum_{(k,j) \in {\mathcal{H}}_{1}} \int _{E_{j}^{k}} m_{Q_{j}^{k}}(f_{2})^{p_{+}(Q_{j}^{k})}w(x)\,\mathrm{d}x +1. \end{aligned}$$

Note that

$$ \frac{1}{\sigma (Q_{j}^{k})} \int _{Q_{j}^{k}}f_{2}(y)\sigma (y)^{-1} \sigma (y)\,\mathrm{d}y \le 1 $$

from the definition of $f_{2}$. We calculate

$$\begin{aligned} {\mathrm{I}}_{2,{\mathcal{H}}_{1}}\lesssim{}& \sum_{(k,j) \in {\mathcal{H}}_{1}} \int _{E_{j}^{k}} \biggl( \frac{1}{\sigma (Q_{j}^{k})} \int _{Q_{j}^{k}}f_{2}(y) \sigma (y)^{-1} \sigma (y)\,\mathrm{d}y \biggr)^{p_{+}(Q_{j}^{k})} \biggl( \frac{\sigma (Q_{j}^{k})}{ \vert Q_{j}^{k} \vert } \biggr)^{p_{+}(Q_{j}^{k})}w(x)\,\mathrm{d}x +1 \\ \le{}& \sum_{(k,j) \in {\mathcal{H}}_{1}} \int _{E_{j}^{k}} \biggl( \frac{1}{\sigma (Q_{j}^{k})} \int _{Q_{j}^{k}}f_{2}(y)\sigma (y)^{-1} \sigma (y)\,\mathrm{d}y \biggr)^{p_{\infty }} \biggl( \frac{\sigma (Q_{j}^{k})}{ \vert Q_{j}^{k} \vert } \biggr)^{p_{+}(Q_{j}^{k})}w(x)\,\mathrm{d}x \\ &{} + \sum_{(k,j) \in {\mathcal{H}}_{1}} \int _{E_{j}^{k}}\bigl(e+ \vert x \vert \bigr)^{-K} \biggl(\frac{\sigma (Q_{j}^{k})}{ \vert Q_{j}^{k} \vert } \biggr)^{p_{+}(Q_{j}^{k})}w(x)\,\mathrm{d}x+1. \end{aligned}$$

Since $\sigma (Q_{j}^{k}) \le 1$ and $p(x) \le p_{+}(Q_{j}^{k})$ for $x \in Q_{j}^{k}$, we have

$$\begin{aligned} {\mathrm{I}}_{2,{\mathcal{H}}_{1}}\lesssim{}& \sum_{(k,j) \in {\mathcal{H}}_{1}} \int _{E_{j}^{k}} \biggl( \frac{1}{\sigma (Q_{j}^{k})} \int _{Q_{j}^{k}}f_{2}(y) \sigma (y)^{-1} \sigma (y)\,\mathrm{d}y \biggr)^{p_{\infty }} \biggl( \frac{\sigma (Q_{j}^{k})}{ \vert Q_{j}^{k} \vert } \biggr)^{p_{+}(Q_{j}^{k})}w(x)\,\mathrm{d}x \\ &{} + \sum_{(k,j) \in {\mathcal{H}}_{1}} \int _{E_{j}^{k}}\bigl(e+ \vert x \vert \bigr)^{-K} \frac{\sigma (Q_{j}^{k})^{p_{-}(Q_{j}^{k})}}{ \vert Q_{j}^{k} \vert ^{p(x)}}w(x)\,\mathrm{d}x+1 \\ \lesssim{}& \sum_{(k,j) \in {\mathcal{H}}_{1}} \int _{E_{j}^{k}} \biggl( \frac{1}{\sigma (Q_{j}^{k})} \int _{Q_{j}^{k}}f_{2}(y)\sigma (y)^{-1} \sigma (y)\,\mathrm{d}y \biggr)^{p_{\infty }} \biggl( \frac{\sigma (Q_{j}^{k})}{ \vert Q_{j}^{k} \vert } \biggr)^{p_{+}(Q_{j}^{k})}w(x)\,\mathrm{d}x+1, \end{aligned}$$

by virtue of Lemma 2.20. Consequently, we only have to show

$$ \sum_{(k,j) \in {\mathcal{H}}_{1}} \int _{E_{j}^{k}} \biggl( \frac{1}{\sigma (Q_{j}^{k})} \int _{Q_{j}^{k}}f_{2}(y)\sigma (y)^{-1} \sigma (y)\,\mathrm{d}y \biggr)^{p_{\infty }} \biggl( \frac{\sigma (Q_{j}^{k})}{ \vert Q_{j}^{k} \vert } \biggr)^{p_{+}(Q_{j}^{k})}w(x)\,\mathrm{d}x \lesssim 1. $$

In fact, from Lemma 2.19(3), we get

$$\begin{aligned} &\sum_{(k,j) \in {\mathcal{H}}_{1}} \int _{E_{j}^{k}} \biggl( \frac{1}{\sigma (Q_{j}^{k})} \int _{Q_{j}^{k}}f_{2}(y)\sigma (y)^{-1} \sigma (y)\,\mathrm{d}y \biggr)^{p_{\infty }} \frac{\sigma (Q_{j}^{k})^{p_{-}(Q_{j}^{k})}}{ \vert Q_{j}^{k} \vert ^{p(x)}}w(x)\,\mathrm{d}x \\ &\quad \le \sum_{(k,j) \in {\mathcal{H}}_{1}} \biggl( \frac{1}{\sigma (Q_{j}^{k})} \int _{Q_{j}^{k}}f_{2}(y)\sigma (y)^{-1} \sigma (y)\,\mathrm{d}y \biggr)^{p_{\infty }} \sigma \bigl(Q_{j}^{k} \bigr). \end{aligned}$$

Thus, using Lemmas 2.18 and 2.20, we have

$$\begin{aligned}& \sum_{(k,j) \in {\mathcal{H}}_{1}} \int _{E_{j}^{k}} \biggl( \frac{1}{\sigma (Q_{j}^{k})} \int _{Q_{j}^{k}}f_{2}(y)\sigma (y)^{-1} \sigma (y)\,\mathrm{d}y \biggr)^{p_{\infty }} \frac{\sigma (Q_{j}^{k})^{p_{-}(Q_{j}^{k})}}{ \vert Q_{j}^{k} \vert ^{p(x)}}w(x)\,\mathrm{d}x \\& \quad \lesssim \sum_{(k,j) \in {\mathcal{H}}_{1}} \biggl( \frac{1}{\sigma (Q_{j}^{k})} \int _{Q_{j}^{k}}f_{2}(y)\sigma (y)^{-1} \sigma (y)\,\mathrm{d}y \biggr)^{p_{\infty }} \sigma \bigl(E^{k}_{j} \bigr) \\& \quad \lesssim \int _{{\mathbb{R}}^{n}}\bigl(f_{2}(y)\sigma (y)^{-1}\bigr)^{p_{\infty }} \sigma (x)\,\mathrm{d}x \\& \quad \lesssim \int _{{\mathbb{R}}^{n}}\bigl(f_{2}(y)\sigma (y)^{-1}\bigr)^{p(x)} \sigma (x)\,\mathrm{d}x + \int _{{\mathbb{R}}^{n}} \frac{\sigma (x)\,\mathrm{d}x}{(e+ \vert x \vert )^{M}} \lesssim 1. \end{aligned}$$

We consider ${\mathcal{H}}_{2}$. By Hölder’s inequality,

$$ \int _{Q_{j}^{k}}f_{2}(y)\,\mathrm{d}y \lesssim \Vert f_{2} \Vert _{L^{p(\cdot )}(w)} \Vert \chi _{Q_{j}^{k}} \Vert _{L^{p'(\cdot )}(\sigma )} \le \Vert \chi _{Q_{j}^{k}} \Vert _{L^{p'(\cdot )}(\sigma )}. $$

Consequently, using Lemma 2.7 for the measure $w(x) \,\mathrm{d}x$ and Lemma 2.21, we have

$$\begin{aligned} &\sum_{(k,j) \in {\mathcal{H}}_{2}} \int _{E_{j}^{k}} \biggl( \frac{1}{ \vert Q_{j}^{k} \vert } \int _{Q_{j}^{k}}f_{2}(y)\,\mathrm{d}y \biggr)^{p(x)}w(x)\,\mathrm{d}x \\ &\quad =\sum_{(k,j) \in {\mathcal{H}}_{2}} \int _{E_{j}^{k}} \biggl( \frac{1}{ \Vert \chi _{Q_{j}^{k}} \Vert _{L^{p'(\cdot )}(\sigma )}} \int _{Q_{j}^{k}}f_{2}(y)\,\mathrm{d}y \biggr)^{p(x)} \biggl( \frac{ \Vert \chi _{Q_{j}^{k}} \Vert _{L^{p'(\cdot )}(\sigma )}}{ \vert Q_{j}^{k} \vert } \biggr)^{p(x)}w(x)\,\mathrm{d}x \\ &\quad \lesssim \sum_{(k,j) \in {\mathcal{H}}_{2}} \int _{E_{j}^{k}} \biggl( \frac{1}{ \Vert \chi _{Q_{j}^{k}} \Vert _{L^{p'(\cdot )}(\sigma )}} \int _{Q_{j}^{k}}f_{2}(y)\,\mathrm{d}y \biggr)^{p_{\infty }} \biggl( \frac{ \Vert \chi _{Q_{j}^{k}} \Vert _{L^{p'(\cdot )}(\sigma )}}{ \vert Q_{j}^{k} \vert } \biggr)^{p(x)}w(x)\,\mathrm{d}x \\ &\qquad {}+\sum_{(k,j) \in {\mathcal{H}}_{2}} \int _{E_{j}^{k}} \biggl( \frac{ \Vert \chi _{Q_{j}^{k}} \Vert _{L^{p'(\cdot )}(\sigma )}}{ \vert Q_{j}^{k} \vert } \biggr)^{p(x)}\frac{w(x)\,\mathrm{d}x}{(e+ \vert x \vert )^{K}} \\ &\quad \lesssim \sum_{(k,j) \in {\mathcal{H}}_{2}} \int _{E_{j}^{k}} \biggl( \frac{1}{ \Vert \chi _{Q_{j}^{k}} \Vert _{L^{p'(\cdot )}(\sigma )}} \int _{Q_{j}^{k}}f_{2}(y)\,\mathrm{d}y \biggr)^{p_{\infty }} \biggl( \frac{ \Vert \chi _{Q_{j}^{k}} \Vert _{L^{p'(\cdot )}(\sigma )}}{ \vert Q_{j}^{k} \vert } \biggr)^{p(x)}w(x)\,\mathrm{d}x+1 \\ &\quad \equiv J_{1}+1. \end{aligned}$$

Thanks to Lemma 2.15 applied to the dual exponent,

$$ \biggl( \frac{\sigma (Q_{j}^{k})}{ \Vert \chi _{Q_{j}^{k}} \Vert _{L^{p'(\cdot )}(\sigma )}} \biggr)^{p_{\infty }} \lesssim \bigl(\sigma \bigl(Q_{j}^{k}\bigr)^{1- \frac{1}{p'_{\infty }}} \bigr)^{p_{\infty }} = \sigma \bigl(Q_{j}^{k}\bigr). $$

Thus we obtain

$$\begin{aligned} J_{1} &= \sum_{(k,j) \in {\mathcal{H}}_{2}} \int _{E_{j}^{k}} \biggl( \frac{1}{\sigma (Q_{j}^{k})} \int _{Q_{j}^{k}}f_{2}(y)\,\mathrm{d}y \biggr)^{p_{\infty }} \biggl( \frac{\sigma (Q_{j}^{k})}{ \Vert \chi _{Q_{j}^{k}} \Vert _{L^{p'(\cdot )}(\sigma )}} \biggr)^{p_{\infty }} \\ &\quad \times \biggl( \frac{ \Vert \chi _{Q_{j}^{k}} \Vert _{L^{p'(\cdot )}(\sigma )}}{ \vert Q_{j}^{k} \vert } \biggr)^{p(x)}w(x)\,\mathrm{d}x \\ &\le \sum_{(k,j) \in {\mathcal{H}}_{2}} \int _{E_{j}^{k}} \biggl( \frac{1}{\sigma (Q_{j}^{k})} \int _{Q_{j}^{k}}f_{2}(y)\,\mathrm{d}y \biggr)^{p_{\infty }} \biggl( \frac{ \Vert \chi _{Q_{j}^{k}} \Vert _{L^{p'(\cdot )}(\sigma )}}{ \vert Q_{j}^{k} \vert } \biggr)^{p(x)} \sigma \bigl(Q_{j}^{k}\bigr) w(x)\,\mathrm{d}x \\ &\lesssim \sum_{(k,j) \in {\mathcal{H}}_{2}} \biggl( \frac{1}{\sigma (Q_{j}^{k})} \int _{Q_{j}^{k}}f_{2}(y)\,\mathrm{d}y \biggr)^{p_{\infty }} \sigma \bigl(Q_{j}^{k}\bigr) \\ &\lesssim 1, \end{aligned}$$

where in the third inequality, we used (2.7) and Lemmas 2.7 and 2.9. Together, we obtain the desired result.

3.3 Equivalent condition on weights

Finally, we consider the condition on which Q in the definition of $w \in A^{\mathrm{loc}}_{p(\cdot )}$. We generalize $w \in A^{\mathrm{loc}}_{p(\cdot )}$ as follows:

Definition 3.1

Given an exponent $p(\cdot ):\mathbb{R}^{n} \to (1,\infty )$ with $p_{-}>1$, a positive number $R \ge 1$ and a weight w, we say that $w \in A^{{\mathrm{loc}},R}_{p(\cdot )}$ if

$$ [w]_{A^{{\mathrm{loc}},R}_{p(\cdot )}} \equiv \sup_{ \vert Q \vert \le R^{n}} \vert Q \vert ^{-1} \Vert \chi _{Q} \Vert _{L^{p(\cdot )}(w)} \Vert \chi _{Q} \Vert _{L^{p'(\cdot )}(\sigma )}< \infty , $$

where $\sigma \equiv w^{-\frac{1}{p(\cdot )-1}}$ as before and the supremum is taken over all cubes $Q \in \mathcal{Q}$ with volume $R^{n}$.

Accordingly, we consider the local maximal operator given by

$$ M^{{\mathrm{loc}},R} f(x) \equiv \sup_{Q \in \mathcal{Q}, \vert Q \vert \le R^{n}} \frac{\chi _{Q}(x)}{ \vert Q \vert } \int _{Q} \bigl\vert f(y) \bigr\vert {\,\mathrm{d}}y \quad \bigl(x \in { \mathbb{R}}^{n}\bigr) $$

for a measurable function f and $R \ge 1$.

Similar to Theorem 1.2, we can prove the following theorem:

Theorem 3.2

Suppose that a variable exponent $p(\cdot ):\mathbb{R}^{n} \to [1,\infty )$ satisfies conditions (1.1) and (1.2) and $1< p_{-} \le p_{+}<\infty$. Let $R \ge 1$. Then given any $w \in A_{p(\cdot )}^{{\mathrm{loc}},R}$,

$$ \bigl\Vert M^{{\mathrm{loc}},R}f \bigr\Vert _{L^{p(\cdot )}(w)} \le C \Vert f \Vert _{L^{p(\cdot )}(w)}. $$

We remark that the class $w \in A_{p(\cdot )}^{{\mathrm{loc}},R}$ with $R \ge 1$ is independent of $R \ge 1$.

Proposition 3.3

Suppose that a variable exponent $p(\cdot ):\mathbb{R}^{n} \to [1,\infty )$ satisfies conditions (1.1) and (1.2) and $1< p_{-} \le p_{+}<\infty$. The class $w \in A_{p(\cdot )}^{{\mathrm{loc}},R}$ with $R \ge 1$ is independent of $R \ge 1$.

Proof

Let $w \in A_{p(\cdot )}^{\mathrm{loc}}$. If $m\equiv [2R+1]$, then $M^{{\mathrm{loc}},R}f(x) \le C_{m} (M^{\mathrm{loc}})^{m} f(x)$ for any measurable function f, where $(M^{\mathrm{loc}})^{m}$ denotes the m-fold composition of $M^{\mathrm{loc}}$. Consequently, $M^{{\mathrm{loc}},R}$ is bounded on $L^{p(\cdot )}(w)$. Thus, $w \in A_{p(\cdot )}^{{\mathrm{loc}},R}$. □

4 Proof of Theorem 1.5

Thanks to Lemma 2.8, we have

$$ \bigl\Vert M^{\mathrm{loc}}_{{\mathfrak{D}}}f \bigr\Vert _{L^{p(\cdot )}(w)} \sim \biggl( \sum_{Q \in {\mathcal{D}}_{0,(1,1,\ldots ,1)}} \bigl( \bigl\Vert \bigl(M^{\mathrm{loc}}_{{ \mathfrak{D}}}f\bigr)\chi _{Q} \bigr\Vert _{L^{p(\cdot )}(w)} \bigr)^{p_{\infty }} \biggr)^{\frac{1}{p_{\infty }}}. $$

Since $(M^{\mathrm{loc}}_{{\mathfrak{D}}}f)\chi _{Q} =M^{\mathrm{loc}}_{{\mathfrak{D}}}[f \chi _{Q}] =(M_{{\mathfrak{D}}}[f\chi _{Q}])\chi _{Q}$ for any $Q \in {\mathcal{D}}_{0,(1,1,\ldots ,1)}$, we can use Theorem 1.7 to get

$$ \bigl\Vert M^{\mathrm{loc}}_{{\mathfrak{D}}}f \bigr\Vert _{L^{p(\cdot )}(w)} \lesssim \biggl( \sum_{Q \in {\mathcal{D}}_{0,(1,1,\ldots ,1)}} \bigl( \Vert f\chi _{Q} \Vert _{L^{p( \cdot )}(w)} \bigr)^{p_{\infty }} \biggr)^{\frac{1}{p_{\infty }}}. $$

Using Lemma 2.8, we obtain the desired result.

5 Application – the weighted vector-valued maximal inequality

Finally, as an application, we consider the weighted vector-valued inequality for $M^{\mathrm{loc}}$. If Theorem 1.8 is proven, Theorem 1.11 follows immediately from Proposition 1.9. So, we concentrate on Theorem 1.8 using an extrapolation for $A^{\mathrm{loc}}_{p(\cdot )}$. We prepare two lemmas.

Lemma 5.1

Let $w_{0}, w_{1} \in A^{\mathrm{loc}}_{1}$ and $1< p<\infty$. Then, $w=w_{0}w_{1}^{1-p} \in A^{\mathrm{loc}}_{p}$.

Proof

The proof is analogous to the corresponding assertion for $A_{1}$ and $A_{p}$. For convenience, here we supply the proof. Fix a cube Q with $|Q|\le 1$. Then

$$\begin{aligned} &\frac{1}{ \vert Q \vert } \int _{Q} w_{0}(x)w_{1}(x)^{1-p} {\,\mathrm{d}}x \biggl( \frac{1}{ \vert Q \vert } \int _{Q} w_{0}(x)^{1-p'}w_{1}(x) {\,\mathrm{d}}x \biggr)^{p-1} \\ &\quad \lesssim \frac{1}{ \vert Q \vert } \int _{Q} w_{0}(x) \biggl( \frac{1}{ \vert Q \vert } \int _{Q} w_{1}(y) {\,\mathrm{d}}y \biggr)^{1-p} {\,\mathrm{d}}x \\ &\qquad {}\times \biggl( \frac{1}{ \vert Q \vert } \int _{Q} w_{1}(x) {\,\mathrm{d}}x \biggr)^{p-1} \biggl( \frac{1}{ \vert Q \vert } \int _{Q} w_{0}(y){\,\mathrm{d}}y \biggr)^{-1} \\ &\quad =1. \end{aligned}$$

Thus, $[w]_{A^{\mathrm{loc}}_{p}} \lesssim 1$ and $w \in A^{\mathrm{loc}}_{p}$. □

Let us conclude the proof of Theorem 1.8.

Let $w \in A^{\mathrm{loc}}_{p(\cdot )}$ and $(f, g) \in \mathcal{F}$ with $\|f\|_{L^{p(\cdot )}(w)}<\infty$. Assume that $\|f\|_{L^{p(\cdot )}(w)}>0$ and $0<\|g\|_{L^{p(\cdot )}(w)}<\infty$, also set

$$ h_{1} \equiv \frac{f}{ \Vert f \Vert _{L^{p(\cdot )}(w)}}+ \frac{g}{ \Vert g \Vert _{L^{p(\cdot )}(w)}}. $$

Then, $h_{1} \in L^{p(\cdot )}(w)$ and $\|h_{1}\|_{L^{p(\cdot )}(w)} \le 2$. We define the operator $\mathcal{R}$ as

$$ \mathcal{R}h(x) \equiv \sum_{k=0}^{\infty } \frac{ (M^{\mathrm{loc}} )^{k}h(x)}{2^{k} \Vert M^{\mathrm{loc}} \Vert ^{k}_{\mathcal{B}(L^{p(\cdot )}(w))}} \quad \bigl(x \in \mathbb{R}^{n}\bigr) $$

for $h \in L^{p(\cdot )}(w)$. Then, we can show that

• (i) for all $x \in \mathbb{R}^{n}$, $|h(x)|\le \mathcal{R}h(x)$,

• (ii) $\|\mathcal{R}h\|_{L^{p(\cdot )}(w)} \le 2 \|h\|_{L^{p(\cdot )}(w)}$,

• (iii) $\mathcal{R}h \in A^{\mathrm{loc}}_{1}$ with $[\mathcal{R}h]_{A^{\mathrm{loc}}_{1}} \le 2\|M^{\mathrm{loc}}\|_{\mathcal{B}(L^{p( \cdot )}(w))}$.

Define $M'h \equiv w^{-1}\cdot M^{\mathrm{loc}}(hw)$. Note that if $\sigma =w^{-\frac{1}{p(\cdot )-1}} \in A^{\mathrm{loc}}_{p(\cdot )}$, then $M^{\mathrm{loc}}$ is bounded on $L^{p'(\cdot )}(\sigma )$. Hence, $M'$ is bounded on $L^{p'(\cdot )}(w)$. In fact,

$$\begin{aligned} \bigl\Vert M'h \bigr\Vert _{L^{p'(\cdot )}(w)} &= \bigl\Vert w^{-1}\cdot M^{\mathrm{loc}}(hw) w^{ \frac{1}{p'(\cdot )}} \bigr\Vert _{L^{p'(\cdot )}} \\ &= \bigl\Vert M^{\mathrm{loc}}(hw) \cdot \sigma ^{\frac{1}{p'(\cdot )}} \bigr\Vert _{L^{p'( \cdot )}} \\ &= \bigl\Vert M^{\mathrm{loc}}(hw) \bigr\Vert _{L^{p'(\cdot )}(\sigma )} \lesssim \Vert hw \Vert _{L^{p'( \cdot )}(\sigma )} = \Vert h \Vert _{L^{p'(\cdot )}(w)}. \end{aligned}$$

Moreover, define

$$ \mathcal{R}'h(x) \equiv \sum_{k=0}^{\infty } \frac{ (M' )^{k}h(x)}{2^{k} \Vert M' \Vert ^{k}_{\mathcal{B}(L^{p'(\cdot )}(w))}} \quad \bigl(x \in \mathbb{R}^{n}\bigr) $$

for $h \in L^{p'(\cdot )}$. Then, we also have

• (i) for all $x \in \mathbb{R}^{n}$, $|h(x)|\le \mathcal{R'}h(x)$,

• (ii) $\|\mathcal{R'}h\|_{L^{p'(\cdot )}(w)} \le 2 \|h\|_{L^{p'(\cdot )}(w)}$,

• (iii) $(\mathcal{R'}h)w \in A^{\mathrm{loc}}_{1}$ with $[(\mathcal{R}'h)w]_{A^{\mathrm{loc}}_{1}} \le 2\|M'\|_{\mathcal{B}(L^{p'( \cdot )}(w))}$.

Fix $f \in L^{p(\cdot )}(w)$. Then, $fw^{\frac{1}{p(\cdot )}} \in L^{p(\cdot )}$. Thus, by duality there exists a nonnegative function $h \in L^{p'(\cdot )}$ with $\|h\|_{L^{p'(\cdot )}}=1$ such that

$$\begin{aligned} & \Vert f \Vert _{L^{p(\cdot )}(w)} \\ &\quad \lesssim \int _{\mathbb{R}^{n}} f(x)h(x)w(x)^{\frac{1}{p(x)}} {\,\mathrm{d}}x \\ &\quad \le \int _{\mathbb{R}^{n}} f(x)\mathcal{R}h_{1}(x)^{- \frac{1}{p'_{0}}} \mathcal{R}h_{1}(x)^{\frac{1}{p'_{0}}} \mathcal{R}' \bigl[hw^{-\frac{1}{p'(\cdot )}} \bigr](x)^{\frac{1}{p_{0}}} \mathcal{R}' \bigl[hw^{-\frac{1}{p'(\cdot )}} \bigr](x)^{ \frac{1}{p'_{0}}}w(x) {\,\mathrm{d}}x \\ &\quad \le \biggl( \int _{\mathbb{R}^{n}} f(x)^{p_{0}} \mathcal{R}h_{1}(x)^{1-p_{0}} \mathcal{R}' \bigl[hw^{-\frac{1}{p'(\cdot )}} \bigr](x)w(x) { \,\mathrm{d}}x \biggr)^{\frac{1}{p_{0}}} \\ &\qquad {}\times \biggl( \int _{\mathbb{R}^{n}} \mathcal{R}h_{1}(x) \mathcal{R}' \bigl[hw^{-\frac{1}{p'(\cdot )}} \bigr](x)w(x) {\,\mathrm{d}}x \biggr)^{\frac{1}{p'_{0}}} \\ &\quad \equiv I_{1}\times I_{2}. \end{aligned}$$

We estimate $I_{1}$. Since $\mathcal{R}h_{1}, \mathcal{R}' [hw^{-\frac{1}{p'(\cdot )}} ]w \in A^{\mathrm{loc}}_{1}$, according to Lemma 5.1, we have

$$ (\mathcal{R}h_{1} )^{1-p_{0}} \bigl(\mathcal{R}' \bigl[hw^{- \frac{1}{p'(\cdot )}} \bigr]w \bigr) \in A_{p_{0}}. $$

Using our assumption (1.3) and Hölder’s inequality, we have

$$\begin{aligned} I_{1}^{p_{0}} &\le \int _{\mathbb{R}^{n}} g(x)^{p_{0}} \mathcal{R}h_{1}(x)^{1-p_{0}} \mathcal{R}' \bigl[hw^{-\frac{1}{p'(\cdot )}} \bigr](x)w(x) { \,\mathrm{d}}x \\ &\le \int _{\mathbb{R}^{n}} g(x)^{p_{0}} \biggl( \frac{g(x)}{ \Vert g \Vert _{L^{p(\cdot )}(w)}} \biggr)^{1-p_{0}} \mathcal{R}' \bigl[hw^{-\frac{1}{p'(\cdot )}} \bigr](x)w(x) {\,\mathrm{d}}x \\ &= \Vert g \Vert _{L^{p(\cdot )}(w)}^{p_{0}-1} \int _{\mathbb{R}^{n}} g(x) \mathcal{R}' \bigl[hw^{-\frac{1}{p'(\cdot )}} \bigr](x)w(x) {\,\mathrm{d}}x \\ &= \Vert g \Vert _{L^{p(\cdot )}(w)}^{p_{0}-1} \Vert g \Vert _{L^{p(\cdot )}(w)} \bigl\Vert \mathcal{R}' \bigl[hw^{-\frac{1}{p'(\cdot )}} \bigr] \bigr\Vert _{L^{p'(\cdot )}(w)} \\ &\lesssim \Vert g \Vert _{L^{p(\cdot )}(w)}^{p_{0}} \bigl\Vert hw^{- \frac{1}{p'(\cdot )}} \bigr\Vert _{L^{p'(\cdot )}(w)} \\ &= \Vert g \Vert _{L^{p(\cdot )}(w)}^{p_{0}}. \end{aligned}$$

Next, we estimate $I_{2}$. Using Hölder’s inequality, we have

$$\begin{aligned} I_{2}^{p'_{0}} &\lesssim \bigl\Vert ( \mathcal{R}h_{1})w^{ \frac{1}{p(\cdot )}} \bigr\Vert _{L^{p(\cdot )}} \bigl\Vert {\mathcal{R}}'\bigl[hw^{- \frac{1}{p'(\cdot )}}\bigr]\cdot w^{\frac{1}{p'(\cdot )}} \bigr\Vert _{L^{p'( \cdot )}} \\ &= \Vert \mathcal{R}h_{1} \Vert _{L^{p(\cdot )}(w)} \bigl\Vert { \mathcal{R}}'\bigl[hw^{-\frac{1}{p'(\cdot )}}\bigr] \bigr\Vert _{L^{p'(\cdot )}(w)} \\ &\lesssim \Vert h_{1} \Vert _{L^{p(\cdot )}(w)} \bigl\Vert hw^{- \frac{1}{p'(\cdot )}} \bigr\Vert _{L^{p'(\cdot )}(w)} \\ &= \Vert h_{1} \Vert _{L^{p(\cdot )}(w)} \Vert h \Vert _{L^{p'( \cdot )}} \sim 1. \end{aligned}$$

Combining these two estimates gives the desired result.