Boundedness of the Vector-Valued Intrinsic Square Functions on Variable Exponents Herz Spaces

Abstract

In this article, the authors study the boundedness of the vector-valued inequality for the intrinsic square function and the boundedness of the scalar-valued intrinsic square function on variable exponents Herz spaces ${\dot{K}}_{\rho (·)}^{\alpha ,q(·)}\left({\mathbb{R}}^n\right)$. In addition, the boundedness of commutators generated by the scalar-valued intrinsic square function and BMO class is also studied on ${\dot{K}}_{\rho (·)}^{\alpha ,q(·)}\left({\mathbb{R}}^n\right)$.

1. Introduction

Before adequately introducing the subject of this article, it should be noted that our primary goal is to study the boundedness of the vector-valued operators on variable exponent Herz spaces ${\dot{K}}_{p(·)}^{\alpha ,q(·)}\left({\mathbb{R}}^n\right)$. Our main results apply to the intrinsic square function and sublinear operators. In addition, the boundedness of the scalar-valued intrinsic square function and its commutators are also discussed in the aforementioned Herz spaces.

Wilson was the first to propose the intrinsic square functions in [1,2]. In [2], the author obtained the boundedness of the intrinsic square functions on $L^p\left(\omega \right)$ spaces. Since then, numerous researchers have paid a great deal of attention to the study of the boundedness of the vector-valued and scalar-valued intrinsic square functions in various functions spaces. For instance, in [3], the author proved sharp weighted norm inequalities for the intrinsic square function in terms of the $A_p$ (Muckenhoupt weight class) characteristic of $\omega$ for all $p\in (1,\infty )$. In a series of papers by Wang [4,5,6,7], he has proposed the boundedness of the scalar-valued intrinsic square functions on weighted Hardy spaces and weighted Morrey spaces, respectively. Liang et al. [8] discovered the boundedness of intrinsic Littlewood–Paley functions on Musielak–Orlicz Morrey and Campanato spaces. Moreover, we refer to Wang [9], who studied the boundedness of the vector-valued intrinsic square functions on weighted Morrey-type spaces. More applications of such intrinsic square functions can be found in [10,11,12,13,14,15].

The classical notion of Herz spaces ${\dot{K}}_p^{\alpha ,q}\left({\mathbb{R}}^n\right)$ was originally introduced by Herz in [16]. These spaces were extended and studied by many authors [17,18,19,20,21]. The topic of variable exponents function spaces is currently a very active research area (see, for example [22,23,24,25,26,27], and so on), in part due to the breadth of their applications (e.g., in fluid dynamics [28] and in differential equations [29,30,31]). Herz spaces with variable exponents have been studied by many authors using different approaches. The Herz spaces ${\dot{K}}_{p(·)}^{\alpha ,q}\left({\mathbb{R}}^n\right)$ with one variable exponent $p(·)$ were defined by Izuki [25], who also established some boundedness results for certain sublinear operators on such spaces. Wang et al. [32] investigated the Herz spaces ${\dot{K}}_{p(·)}^{\alpha ,q(·)}\left({\mathbb{R}}^n\right)$ with two variable exponents $p(·)$ and $p(·)$, as well as obtaining the boundedness results for parameterized Littlewood–Paley operators on ${\dot{K}}_{p(·)}^{\alpha ,q(·)}\left({\mathbb{R}}^n\right)$. Yang et al. [33] proved the boundedness of the $\theta$-type Calderón–Zygmund operators and commutators on the homogeneous Herz space with variable exponents ${\dot{K}}_{p(·)}^{\alpha ,q(·)}\left({\mathbb{R}}^n\right)$. Cai et al. [34] established the boundedness for the rough singular integral operators and commutators on ${\dot{K}}_{p(·)}^{\alpha ,q(·)}\left({\mathbb{R}}^n\right)$. Another approach including variable exponents was used in [35] to study Herz spaces.

We also mention that Zhuo et al. [36] established the intrinsic square function characterizations of the variable Hardy spaces. Ho [37] proved the boundedness of the vector-valued intrinsic square function on variable Morrey and Bloke spaces. On variable exponent function spaces, Wang [38] investigated the boundedness of commutator of intrinsic square function. For some recent developments, we refer to [39,40].

Motivated by [32,37,38], we study the boundedness of the vector-valued intrinsic square function and the boundedness of the scalar-valued intrinsic square function on the variable exponent Herz spaces. The boundedness of the commutator generated by the scalar-valued intrinsic square function and BMO function is also discussed on ${\dot{K}}_{p(·)}^{\alpha ,q(·)}\left({\mathbb{R}}^n\right)$.

In this article, by Q, we denote a cube in ${\mathbb{R}}^n$. If $E\subseteq {\mathbb{R}}^n$, $\left|E\right|$ denotes its Lebesgue measure set in ${\mathbb{R}}^n$ and ${\chi }_E$ its characteristic function. We always denote a positive constant by C, which is not necessarily the same at each occurrence. By $\hslash \lesssim g$, we mean that $\hslash \le Cg$. The expression $\hslash \approx g$ means that $\hslash \lesssim g\lesssim \hslash .$

2. Definitions and Preliminaries

We review some fundamental lemmas and definitions about variable exponent functions spaces.

Definition 1

([22]). Let $p(·):\mathsf{\Sigma }\to [1,\infty )$ be a measurable function. The variable exponent Lebesgue space is defined as

$$L^{p(·)}\left(\mathsf{\Sigma }\right)=\left\{\hslash \mathit{is}\mathit{measurable}:{\int }_{\mathsf{\Sigma }}{\left(\frac{|\hslash (x\left)\right|}{\vartheta }\right)}^{p\left(x\right)}\mathrm{d}x<\infty \mathit{for}\mathit{some}\mathit{constant}\vartheta >0\right\}.$$

The space $L_{\mathrm{loc}}^{p(·)}\left(\mathsf{\Sigma }\right)$ is defined as

$$L_{\mathrm{loc}}^{p(·)}\left(\mathsf{\Sigma }\right):=\left\{\hslash :\hslash \in L^{p(·)}\left(E\right)\mathit{for}\mathit{every}\mathit{compact}\mathit{subsets}E\subset \mathsf{\Sigma }\right\}.$$

$L^{p(·)}\left(\mathsf{\Sigma }\right)$ is a Banach space with norm given below [27]:

$${\parallel \hslash \parallel }_{L^{p(·)}\left(\mathsf{\Sigma }\right)}=inf\left\{\vartheta >0:{\int }_{\mathsf{\Sigma }}{\left(\frac{|\hslash (x\left)\right|}{\vartheta }\right)}^{p\left(x\right)}\mathrm{d}x\le 1\right\}.$$

For brevity, let us set $p_{-}:=p_{\mathsf{\Sigma }}^{-}:=essinf\left\{p\left(x\right):x\in \mathsf{\Sigma }\right\}$, $p_{+}=p_{\mathsf{\Sigma }}^{+}:=ess$$sup\left\{p\left(x\right):x\in \mathsf{\Sigma }\right\}$.

The notations ${\mathcal{P}}^{*}\left(\mathsf{\Sigma }\right)$,${\mathcal{P}}_0^{*}\left(\mathsf{\Sigma }\right)$ and ${\mathcal{B}}^{*}\left(\mathsf{\Sigma }\right)$ are defined, respectively, as follows:

$$\begin{array}{cc}\hfill {\mathcal{P}}^{*}\left(\mathsf{\Sigma }\right)& :=\left\{p(·)\mathrm{is}\mathrm{measurable}:p_{-}>1\mathrm{and}{p}_{+}<+\infty \right\},\hfill \\ \hfill {\mathcal{P}}_0^{*}\left(\mathsf{\Sigma }\right)& :=\left\{p(·)\mathrm{is}\mathrm{measurable}:p_{-}>0\mathrm{and}{p}_{+}<+\infty \right\}\hfill \end{array}$$

and

$$\begin{array}{c}\hfill {\mathcal{B}}^{*}\left(\mathsf{\Sigma }\right):=\left\{p(·)\in {\mathcal{P}}^{*}\left(\mathsf{\Sigma }\right):M_{HL}\mathrm{is}\mathrm{bounded}\mathrm{on}{L}^{p(·)}\left(\mathsf{\Sigma }\right)\right\},\end{array}$$

where $M_{HL}$ is the standard Hardy–Littlewood maximal operator, which can be defined as

$$(M_{HL}\hslash )\left(x\right)=\underset{r>0}{sup}\frac{1}{r^n}{\int }_{B(x,r)}|\hslash \left(y\right)|dy,\hslash \in L_{\mathrm{loc}}^1\left({\mathbb{R}}^n\right).$$

We say $p(·)\in {\mathcal{B}}^{*}\left({\mathbb{R}}^n\right)$ if $p(·)\in {\mathcal{P}}^{*}\left({\mathbb{R}}^n\right)$ and satisfies the following two inequalities:

$$\begin{array}{c}\hfill |p\left(x\right)-p\left(y\right)|\lesssim -\frac{1}{\log \left(\right|x-y\left|\right)},\mathrm{if}|x-y|\le 1/2;\end{array}$$

(1)

$$\begin{array}{c}\hfill |p\left(x\right)-p\left(y\right)|\lesssim \frac{1}{\log (e+|x\left|\right)},\mathrm{if}\left|y\right|\ge \left|x\right|.\end{array}$$

(2)

See [23,24]. In the sequel, the conjugate of $p\left(x\right)$ is given by the symbol $p^{\prime }\left(x\right)$, which means $p^{\prime }\left(x\right)=p\left(x\right)/(p\left(x\right)-1)$.

Before recalling the definition of Herz spaces with two variable exponents, let us present the following notations: for all $l\in \mathbb{Z}$, we denote $B_l:=\{x\in {\mathbb{R}}^n:\left|x\right|\le 2^l\},{\Re }_l:=B_l\backslash B_{l-1}\mathrm{and}{\chi }_l:={\chi }_{R_l}$. Denote ${\mathbb{Z}}^{+}$ as the sets of positive integers, ${\tilde{\chi }}_l:={\chi }_{R_l}$ if $l\in {\mathbb{Z}}^{+}$ and ${\tilde{\chi }}_0:={\chi }_{B_0}$. The notation of the mixed Lebesgue-sequence space ${\ell }^{q(·)}\left(L^{p(·)}\right)$, which is first introduced by Almeida et al. in [41]. Let $p(·)$ and $q(·)\in {\mathcal{P}}_0^{*}\left({\mathbb{R}}^n\right)$. Given a sequence of functions ${\left\{{\hslash }_j\right\}}_{j\in \mathbb{Z}}$, define the modular

$${\wp }_{{\ell }^{q(·)}\left(L^{p(·)}\right)}\left({\left\{{\hslash }_j\right\}}_{j\in \mathbb{Z}}\right):=\sum _{j\in \mathbb{Z}}inf\left\{{\vartheta }_j>0:{\int }_{{\mathbb{R}}^n}{\left(\frac{|{\hslash }_j\left(x\right)|}{{\vartheta }_j^{1/q\left(x\right)}}\right)}^{p\left(x\right)}\mathrm{d}x\le 1\right\}.$$

The quasi-norm is defined as follows:

$$\begin{array}{c}\hfill {∥{\left\{{\hslash }_j\right\}}_{j\in \mathbb{Z}}∥}_{{\ell }^{q(·)}\left(L^{p(·)}\right)}:=inf\left\{\vartheta >0:{\wp }_{{\ell }^{q(·)}\left(L^{p(·)}\right)}\left(\frac{{\left\{{\hslash }_j\right\}}_{j\in \mathbb{Z}}}{\vartheta }\right)\le 1\right\}.\end{array}$$

(3)

When $p(·)\in {\mathcal{P}}_0^{*}\left({\mathbb{R}}^n\right)$, then the space $L^{p(·)}\left({\mathbb{R}}^n\right)$ can be defined as follows:

$$L^{p(·)}\left({\mathbb{R}}^n\right):=\left\{{\hslash :|\hslash |}^{p_0}\in L^{q(·)}\left({\mathbb{R}}^n\right)\right.\mathrm{for}\mathrm{some}{p}_0\mathrm{with}{p}_0\in (0,p_{-})\mathrm{and}\left.q\left(x\right)=p\left(x\right)p_0^{-1}\right\}.$$

and its quasi-norm is defined as follows:

$${\parallel \hslash \parallel }_{L^{p(·)}}:={∥{\left|\hslash \right|}^{p_0}∥}_{L^{q(·)}}^{1/p_0}.$$

Thus, we may substitute (3) for the simple formula

$${\wp }_{{\ell }^{q(·)}\left(L^{p(·)}\right)}\left({\left\{{\hslash }_j\right\}}_{j\in \mathbb{Z}}\right):=\sum _{j\in \mathbb{Z}}{∥{\left|{\hslash }_j\right|}^{q(·)}∥}_{L^{\frac{p(·)}{q(·)}}\left({\mathbb{R}}^n\right)}.$$

Definition 2

([32]). Let $p(·),q(·)\in {\mathcal{P}}_0^{*}\left({\mathbb{R}}^n\right)$ and let $\alpha \in \mathbb{R}$. The homogeneous Herz space ${\dot{K}}_{p(·)}^{\alpha ,q(·)}\left({\mathbb{R}}^n\right)$ is defined as

$${\dot{K}}_{p(·)}^{\alpha ,q(·)}\left({\mathbb{R}}^n\right)=\left\{\hslash \in L_{\mathrm{loc}}^{p(·)}\left({\mathbb{R}}^n\backslash \left\{0\right\}\right):{\parallel \hslash \parallel }_{{\dot{K}}_{p(·)}^{\alpha ,q(·)}\left({\mathbb{R}}^n\right)}<\infty \right\},$$

where

$$\begin{array}{cc}\hfill {\parallel \hslash \parallel }_{{\dot{K}}_{p(·)}^{\alpha ,q(·)}\left({\mathbb{R}}^n\right)}& :={∥{\left\{2^{l\alpha }\left|\hslash {\chi }_l\right|\right\}}_{l=-\infty }^{\infty }∥}_{{\ell }^{q(·)}\left(L^{p(·)}\right)}\hfill \\ & =inf\left\{\vartheta >0:\sum _{l=-\infty }^{\infty }{∥{\left(\frac{2^{l\alpha }\left|\hslash {\chi }_l\right|}{\vartheta }\right)}^{q(·)}∥}_{L^{\frac{p(·)}{q(·)}}}\le 1\right\}.\hfill \end{array}$$

The non-homogeneous Herz space $K_{p(·)}^{\alpha ,q(·)}\left({\mathbb{R}}^n\right)$ is defined as

$$K_{p(·)}^{\alpha ,q(·)}\left({\mathbb{R}}^n\right):=\left\{\hslash \in L_{\mathrm{loc}}^{p(·)}\left({\mathbb{R}}^n\backslash \left\{0\right\}\right):{\parallel \hslash \parallel }_{{\dot{K}}_{p(·)}^{\alpha ,q(·)}\left({\mathbb{R}}^n\right)}<\infty \right\},$$

where

$$\begin{array}{cc}\hfill {\parallel \hslash \parallel }_{K_{p(·)}^{\alpha ,q(·)}\left({\mathbb{R}}^n\right)}& :={∥{\left\{2^{l\alpha }\left|\hslash {\chi }_l\right|\right\}}_{l=0}^{\infty }∥}_{{\ell }^{q(·)}\left(L^{p(·)}\right)}\hfill \\ & =inf\left\{\vartheta >0:\sum _{l=0}^{\infty }{∥{\left(\frac{2^{l\alpha }\left|\hslash {\chi }_l\right|}{\vartheta }\right)}^{q(·)}∥}_{L^{\frac{p(·)}{q(·)}}}\le 1\right\}.\hfill \end{array}$$

Remark 1.

(1) If $q(·)$ is a constant, then ${\dot{K}}_{p(·)}^{\alpha ,q(·)}\left({\mathbb{R}}^n\right)={\dot{K}}_{p(·)}^{\alpha ,q}\left({\mathbb{R}}^n\right)$. If both $p(·),q(·)$ are constant functions, then ${\dot{K}}_{p(·)}^{\alpha ,q(·)}\left({\mathbb{R}}^n\right)={\dot{K}}_p^{\alpha ,q}\left({\mathbb{R}}^n\right)$. Furthermore, if $\alpha =0$ and $p(·)=q(·)$, then ${\dot{K}}_{p(·)}^{\alpha ,q(·)}\left({\mathbb{R}}^n\right)=K_{p(·)}^{\alpha ,q(·)}\left({\mathbb{R}}^n\right)=L^{p(·)}\left({\mathbb{R}}^n\right)$.

(2) If $q_1(·)\mathrm{and}{q}_2(·)\in {\mathcal{P}}_0^{*}\left({\mathbb{R}}^n\right)$ satisfying ${\left(q_1\right)}_{+}\le {\left(q_2\right)}_{-}$, then (see [32])

$${\dot{K}}_{p(·)}^{\alpha ,q_1(·)}\left({\mathbb{R}}^n\right)\subset {\dot{K}}_{p(·)}^{\alpha ,q_2(·)}\left({\mathbb{R}}^n\right).$$

Proposition 1

([34]). Let ${\theta }_l\in [0,\infty ),1\le p_l\le {sup}_{l\in \mathbb{Z}}{p}_l<\infty (l\in \mathbb{Z})$. Then

$$\sum _{l\in \mathbb{Z}}{\theta }_l^{p_l}\le 2·{\left(\sum _{l\in \mathbb{Z}}{\theta }_l\right)}^{p_{*}},$$

where

$$p_{*}=\left\{\begin{array}{c}\underset{l\in \mathbb{Z}}{inf}{p}_{l}if{\displaystyle \sum _{l\in \mathbb{Z}}}{\theta }_l\le 1,\hfill \\ {\displaystyle \underset{l\in \mathbb{Z}}{sup}}{p}_{l}if{\displaystyle \sum _{l\in \mathbb{Z}}}{\theta }_l>1.\hfill \end{array}\right.$$

When $p(·)\in {\mathcal{P}}^{*}\left({\mathbb{R}}^n\right),$ Hölder’s inequality holds true in the following form:

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^n}|\hslash \left(x\right)f\left(x\right)|dx\le \left(1+\frac{1}{p_{-}}-\frac{1}{p_{+}}\right){\parallel \hslash \parallel }_{L^{p(·)}}{\parallel f\parallel }_{L^{p^{\prime }(·)}}.\end{array}$$

(4)

See [22].

Lemma 1

([25]). Let $p(·)\in {\mathcal{B}}^{*}\left({\mathbb{R}}^n\right).$ Then, we have for any ball B in ${\mathbb{R}}^n$,

$${\left|B\right|}^{-1}\parallel {\chi }_B{\parallel }_{L^{p(·)}}{\parallel {\chi }_B\parallel }_{L^{p^{\prime }(·)}}\le C.$$

Lemma 2

([25]). Let $p_{\aleph }(·)\in {\mathcal{B}}^{*}\left({\mathbb{R}}^n\right),\aleph =1,2$. Then, there is a positive constant C such that for every ball B in ${\mathbb{R}}^n$ and any measurable subset $S\subset B,$

$$\frac{\parallel {\chi }_S{\parallel }_{L^{p_{\aleph }(·)}}}{\parallel {\chi }_B{\parallel }_{L^{p_{\aleph }(·)}}}\le C\frac{\left|S\right|}{\left|B\right|},\frac{\parallel {\chi }_S{\parallel }_{L^{p_{\aleph }^{\prime }(·)}}}{\parallel {\chi }_B{\parallel }_{L^{p_{\aleph }^{\prime }(·)}}}\le C{\left(\frac{\left|S\right|}{\left|B\right|}\right)}^{{\delta }_{\aleph 2}},and\frac{\parallel {\chi }_R{\parallel }_{L^{p_{\aleph }(·)}}}{\parallel {\chi }_B{\parallel }_{L^{p_{\aleph }(·)}}}\le C{\left(\frac{\left|S\right|}{\left|B\right|}\right)}^{{\delta }_{\aleph 1}},$$

where ${\delta }_{\aleph 1},{\delta }_{\aleph 2}$ are constants with $0<{\delta }_{\aleph 1},{\delta }_{\aleph 2}<1$.

The following result was studied by Wang et al. in [32].

Lemma 3.

Let $p(·)\mathrm{and}q(·)\in {\mathcal{P}}^{*}\left({\mathbb{R}}^n\right),\hslash \in L^{p(·)q(·)}$. Then we have

$${min(\parallel \hslash \parallel }_{L^{p(·)q(·)}}^{q_{+}}{,\parallel \hslash \parallel }_{L^{p(·)q(·)}}^{q_{-}}{)\le \parallel |\hslash |}^{q(·)}{\parallel }_{L^{p(·)}}\le {max(\parallel \hslash \parallel }_{L^{p(·)q(·)}}^{q_{+}}{,\parallel \hslash \parallel }_{L^{p(·)q(·)}}^{q_{-}}).$$

The following Lemmas are due to [37].

Lemma 4.

Let $0<\beta \le 1$. If $p(·)\in {\mathcal{B}}^{*}\left({\mathbb{R}}^n\right),$ then we have for all $\hslash \in L^{p(·)}$,

$$\begin{array}{c}\hfill {∥S_{\beta }(\hslash )∥}_{L^{p(·)}}\lesssim {∥\hslash ∥}_{L^{p(·)}}.\end{array}$$

(5)

Lemma 5.

Let $r\in (1,\infty ),0<\beta \le 1$. If $p(·)\in {\mathcal{B}}^{*}\left({\mathbb{R}}^n\right),$ then for any ${\left\{{\hslash }_{ı}\right\}}_{ı\in \mathbb{N}}$ satisfying ${∥{\left\{{\hslash }_{ı}\right\}}_{ı\in \mathbb{N}}∥}_{{\ell }^r}\in L^{p(·)}$, we have

$$\begin{array}{c}\hfill {∥{∥{\left\{S_{\beta }\left({\hslash }_{ı}\right)\right\}}_{ı}∥}_{{\ell }^r}∥}_{L^{p(·)}}\lesssim {∥{∥{\left\{{\hslash }_{ı}\right\}}_{ı}∥}_{{\ell }^r}∥}_{L^{p(·)}}.\end{array}$$

(6)

3. Boundedness of Operators on Herz Spaces

3.1. Vector-Valued Intrinsic Square Function

For $0<\beta \le 1$, the family ${\mathcal{C}}_{\beta }$ consists of those functions $\psi :{\mathbb{R}}^n\to (0,\infty )$ such that $supp\psi \subseteq \left\{x\in {\mathbb{R}}^n:\left|x\right|\le 1\right\},{\int }_{{\mathbb{R}}^n}\psi \left(x\right)\mathrm{d}x=0$ and for $x_1,x_2,$

$$\begin{array}{c}\hfill \left|\psi \left(x_1\right)-\psi \left(x_2\right)\right|\le {\left|x_1-x_2\right|}^{\beta }.\end{array}$$

(7)

For any $\hslash \in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1\left({\mathbb{R}}^n\right)$ and $(y,t)\in {\mathbb{R}}_{+}^{n+1}={\mathbb{R}}^n\times (0,\infty )$, let us set

$$\begin{array}{c}\hfill {\mathcal{A}}_{\beta }(y,t):=\underset{\psi \in {\mathcal{C}}_{\beta }}{sup}\left|\left(\hslash \ast {\psi }_t\right)\left(y\right)\right|:=\underset{\psi \in {\mathcal{C}}_{\beta }}{sup}\left|{\int }_{{\mathbb{R}}^n}{\psi }_t(y-z)\hslash \left(z\right)\mathrm{d}z\right|,\end{array}$$

(8)

where ${\psi }_t\left(y\right)=t^{-n}\psi \left(\frac{y}{t}\right)$ and ${\mathcal{A}}_{\beta }$ is Lebesgue measurable.

The intrinsic square function of ℏ of order $\beta$ is defined as follows:

$$\begin{array}{c}\hfill {\mathcal{S}}_{\beta }(\hslash )\left(x\right)={\left({\int \int }_{{\mathsf{\Gamma }}_{}\left(x\right)}{\left({\mathcal{A}}_{\beta }(y,t)\right)}^2\frac{\mathrm{d}t\mathrm{d}y}{t^{1+n}}\right)}^{1/2}\forall x\in {\mathbb{R}}^n,\end{array}$$

(9)

where ${\mathsf{\Gamma }}_{}\left(x\right)=\left\{(y,t)\in {\mathbb{R}}_{+}^{n+1}:|x-y|<t\right\}.$

Theorem 1.

Suppose that $1<r<\infty ,q_1(·)\mathrm{and}{q}_2(·)\in {\mathcal{P}}_0^{*}\left({\mathbb{R}}^n\right)$ with ${\left(q_1\right)}_{+}\le {\left(q_2\right)}_{-}$ and let $\beta \in (0,1],p(·)\in {\mathcal{B}}^{*}\left({\mathbb{R}}^n\right)$. If $\alpha \in (-n{\delta }_{11},n{\delta }_{12})$, where ${\delta }_{11},{\delta }_{12}\in (0,1)$ are constants stated in Lemma 2, then for any ${\left\{{\hslash }_{ı}\right\}}_{ı\in \mathbb{N}}$ with ${∥{∥{\left\{{\hslash }_{ı}\right\}}_{ı}∥}_{{\ell }^r}∥}_{{\dot{K}}_{p(·)}^{\alpha ,q_1(·)}\left({\mathbb{R}}^n\right)}<\infty ,$ the following vector-valued inequality holds:

$${∥{\left(\sum _{ı}{\left|S_{\beta }\left({\hslash }_{ı}\right)\right|}^r\right)}^{\frac{1}{r}}∥}_{{\dot{K}}_{p(·)}^{\alpha ,q_2(·)}}\le C{∥{\left(\sum _{ı}{\left|{\hslash }_{ı}\right|}^r\right)}^{\frac{1}{r}}∥}_{{\dot{K}}_{p(·)}^{\alpha ,q_1(·)}}.$$

Proof. 

For any ${\left\{{\hslash }_{ı}\right\}}_{ı\in \mathbb{N}}$ satisfies ${∥{∥{\left\{{\hslash }_{ı}\right\}}_{ı}∥}_{{\ell }^r}∥}_{{\dot{K}}_{p(·)}^{\alpha ,q_1(·)}}<\infty ,$ we can write ${\hslash }_{ı}={\sum }_{j=-\infty }^{\infty }{\hslash }_{ı}{\chi }_j={\sum }_{j=-\infty }^{\infty }{\hslash }_{ı}^j.$ By the definition of ${\dot{K}}_{p(·)}^{\alpha ,q(·)}\left({\mathbb{R}}^n\right)$, we have

$${∥{∥{\left\{S_{\beta }\left({\hslash }_{ı}\right)\right\}}_{ı}∥}_{{\ell }^r}{\chi }_l∥}_{{\dot{K}}_{p(·)}^{\alpha ,q_2(·)}\left({\mathbb{R}}^n\right)}=inf\left\{\vartheta >0:\sum _{l=-\infty }^{\infty }{∥{\left(\frac{2^{l\alpha }{∥{\left\{S_{\beta }\left({\hslash }_{ı}\right)\right\}}_{ı}∥}_{{\ell }^r}{\chi }_l}{\vartheta }\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}\le 1\right\}$$

Since

$$\begin{array}{cc}\hfill {∥{\left(\frac{2^{l\alpha }{∥{\left\{S_{\beta }\left({\hslash }_{ı}\right)\right\}}_{ı}∥}_{{\ell }^r}{\chi }_l}{\vartheta }\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}\le & {∥{\left(\frac{2^{l\alpha }{∥{\left\{{\sum }_{j=-\infty }^{\infty }{S}_{\beta }\left({\hslash }_{ı}^j\right)\right\}}_{ı}∥}_{{\ell }^r}{\chi }_l}{{\vartheta }_{11}+{\vartheta }_{12}+{\vartheta }_{13}}\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}\hfill \end{array}$$

$$\begin{array}{cc}\hfill \le & C{∥{\left(\frac{2^{l\alpha }{∥{\left\{{\sum }_{j=-\infty }^{l-3}{S}_{\beta }\left({\hslash }_{ı}^j\right)\right\}}_{ı}∥}_{{\ell }^r}{\chi }_l}{{\vartheta }_{11}}\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}\hfill \\ & +C{∥{\left(\frac{2^{l\alpha }{∥{\left\{{\sum }_{j=l-2}^{l+2}{S}_{\beta }\left({\hslash }_{ı}^j\right)\right\}}_{ı}∥}_{{\ell }^r}{\chi }_l}{{\vartheta }_{12}}\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}\hfill \\ & +C{∥{\left(\frac{2^{l\alpha }{∥{\left\{{\sum }_{j=l+3}^{\infty }{S}_{\beta }\left({\hslash }_{ı}^j\right)\right\}}_{ı}∥}_{{\ell }^r}{\chi }_l}{{\vartheta }_{13}}\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}},\hfill \end{array}$$

where

$$\begin{array}{cc}& {\vartheta }_{11}={∥{\left\{2^{l\alpha }{∥{\left\{\sum _{j=-\infty }^{l-3}{S}_{\beta }\left({\hslash }_{ı}^j\right)\right\}}_{ı}∥}_{{\ell }^r}{\chi }_l\right\}}_{l=-\infty }^{\infty }∥}_{{\ell }^{q_2(·)}\left(L^{p(·)}\right)},\hfill \\ & {\vartheta }_{12}={∥{\left\{2^{l\alpha }{∥{\left\{\sum _{j=l-2}^{l+2}{S}_{\beta }\left({\hslash }_{ı}^j\right)\right\}}_{ı}∥}_{{\ell }^r}{\chi }_l\right\}}_{l=-\infty }^{\infty }∥}_{{\ell }^{q_2(·)}\left(L^{p(·)}\right)},\hfill \\ & {\vartheta }_{13}={∥{\left\{2^{l\alpha }{∥{\left\{\sum _{j=l+3}^{\infty }{S}_{\beta }\left({\hslash }_{ı}^j\right)\right\}}_{ı}∥}_{{\ell }^r}{\chi }_l\right\}}_{l=-\infty }^{\infty }∥}_{{\ell }^{q_2(·)}\left(L^{p(·)}\right)}.\hfill \end{array}$$

If $\vartheta :=({\vartheta }_{11}+{\vartheta }_{12}+{\vartheta }_{13}):={\sum }_{\varrho =1}^3{\vartheta }_{1\varrho }$, thus,

$$\sum _{l=-\infty }^{\infty }{∥{\left(\frac{2^{l\alpha }{∥{\left\{S_{\beta }\left({\hslash }_{ı}\right)\right\}}_{ı}∥}_{{\ell }^r}{\chi }_l}{\vartheta }\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}\le C.$$

Then,

$${∥{∥{\left\{S_{\beta }\left({\hslash }_{ı}\right)\right\}}_{ı}∥}_{{\ell }^r}∥}_{{\dot{K}}_{p(·)}^{\alpha ,q_2(·)}\left({\mathbb{R}}^n\right)}\le C\vartheta :=C\left(\sum _{\varrho =1}^3{\vartheta }_{1\varrho }\right).$$

Hence, if we prove that

$${\vartheta }_{11}\le C{∥{∥{\left\{{\hslash }_{ı}\right\}}_{ı}∥}_{{\ell }^r}∥}_{{\dot{K}}_{p(·)}^{\alpha ,q_1(·)}\left({\mathbb{R}}^n\right)},{\vartheta }_{12}\le C{∥{∥{\left\{{\hslash }_{ı}\right\}}_{ı}∥}_{{\ell }^r}∥}_{{\dot{K}}_{p(·)}^{\alpha ,q_1(·)}\left({\mathbb{R}}^n\right)},{\vartheta }_{13}\le C{∥{∥{\left\{{\hslash }_{ı}\right\}}_{ı}∥}_{{\ell }^r}∥}_{{\dot{K}}_{p(·)}^{\alpha ,q_1(·)}\left({\mathbb{R}}^n\right)},$$

we are done. Let ${\vartheta }_0={\parallel {∥{\left\{{\hslash }_{ı}\right\}}_{ı}∥}_{{\ell }^r}\parallel }_{{\dot{K}}_{p(·)}^{\alpha ,q_1(·)}\left({\mathbb{R}}^n\right)}$.

First, we consider ${\vartheta }_{12}$. For each $l\in \mathbb{Z}$, we define

$${\left(q_2^1\right)}_l=\left\{\begin{array}{c}{\left(q_2\right)}_{-}\mathrm{if}{∥\frac{2^{l\alpha }{∥{\sum }_{j=l-2}^{l+2}{\left\{S_{\beta }\left({\hslash }_{ı}^j\right)\right\}}_{ı}∥}_{{\ell }^r}{\chi }_l}{{\vartheta }_0}∥}_{L^{p(·)}}\le 1,\hfill \\ {\left(q_2\right)}_{+}\mathrm{otherwise}.\hfill \end{array}\right.$$

Using Lemmas 3 and 5, it follows that

$$\begin{array}{cc}& \sum _{l=-\infty }^{\infty }{∥{\left(\frac{2^{l\alpha }{∥{\left\{{\sum }_{j=l-2}^{l+2}{S}_{\beta }\left({\hslash }_{ı}^j\right)\right\}}_{ı}∥}_{{\ell }^r}{\chi }_l}{{\vartheta }_0}\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}\hfill \end{array}$$

$$\begin{array}{cc}& \le C\sum _{l=-\infty }^{\infty }{∥\frac{2^{l\alpha }{∥{\left\{{\sum }_{j=l-2}^{l+2}{S}_{\beta }\left({\hslash }_{ı}^j\right)\right\}}_{ı}∥}_{{\ell }^r}{\chi }_l}{{\vartheta }_0}∥}_{L^{p(·)}}^{{\left(q_2^1\right)}_l}\hfill \\ & \le C\sum _{l=-\infty }^{\infty }{\left(\sum _{j=l-2}^{l+2}{∥\frac{2^{l\alpha }{∥{\left\{S_{\beta }\left({\hslash }_{ı}^j\right)\right\}}_{ı}∥}_{{\ell }^r}{\chi }_l}{{\vartheta }_0}∥}_{L^{p(·)}}\right)}^{{\left(q_2^1\right)}_l}\hfill \\ & \le C\sum _{l=-\infty }^{\infty }{\left(\sum _{j=l-2}^{l+2}{∥{∥{\left\{\frac{D_1{2}^{l\alpha }{\hslash }_{ı}^j}{{\vartheta }_0}\right\}}_{ı}∥}_{{\ell }^r}∥}_{L^{p(·)}}\right)}^{{\left(q_2^1\right)}_l}\hfill \\ & \le C\sum _{l=-\infty }^{\infty }{\left(\sum _{j=l-2}^{l+2}{∥{∥{\left\{\frac{2^{l\alpha }{\hslash }_{ı}^j}{{\vartheta }_0}\right\}}_{ı}∥}_{{\ell }^r}∥}_{L^{p(·)}}\right)}^{{\left(q_2^1\right)}_l},\hfill \end{array}$$

where $C:=max\left\{D_1^{{\left(q_2\right)}_{-}},D_1^{{\left(q_2\right)}_{+}}\right\}$. Since ${∥{\left\{{\hslash }_{ı}\right\}}_{ı}∥}_{{\ell }^r}\in {\dot{K}}_{p(·)}^{\alpha ,q_1(·)}\left({\mathbb{R}}^n\right)$, it is not difficult for us to get ${\displaystyle \sum _{l=-\infty }^{\infty }}{∥{\left(\frac{2^{l\alpha }{∥{\left\{{\hslash }_{ı}^{}\right\}}_{ı}∥}_{{\ell }^r}{\chi }_l}{{\vartheta }_0}\right)}^{q_1(·)}∥}_{L^{\frac{p(·)}{q_1(·)}}}\le 1$ and ${∥\frac{2^{l\alpha }{∥{\left\{{\hslash }_{ı}\right\}}_{ı}∥}_{{\ell }^r}{\chi }_l}{{\vartheta }_0}∥}_{L^{p(·)}}\le 1$. Hence, from Lemma 3 and Proposition 1, we deduce

$$\begin{array}{cc}& \sum _{l=-\infty }^{\infty }{∥{\left(\frac{2^{l\alpha }{∥{\left\{{\sum }_{j=l-2}^{l+2}{S}_{\beta }\left({\hslash }_{ı}^j\right)\right\}}_{ı}∥}_{{\ell }^r}{\chi }_l}{\vartheta }\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}\hfill \\ & \le C\sum _{l=-\infty }^{\infty }{∥{\left(2^{l\alpha }{∥{\left\{\frac{{\hslash }_{ı}}{{\vartheta }_0}\right\}}_{ı}∥}_{{\ell }^r}{\chi }_l\right)}^{q_1(·)}∥}_{L^{\frac{p(·)}{q_1(·)}}}^{\frac{{\left(q_2^1\right)}_l}{(q_1{)}_{+}}}\hfill \\ & \le C{\left\{\sum _{l=-\infty }^{\infty }{∥{\left(2^{l\alpha }{∥{\left\{\frac{{\hslash }_{ı}}{{\vartheta }_0}\right\}}_{ı}∥}_{{\ell }^r}{\chi }_l\right)}^{q_1(·)}∥}_{L^{\frac{p(·)}{q_1(·)}}}\right\}}^{q_{*}^1}\le C,\hfill \end{array}$$

where ${\left(q_2^1\right)}_l\ge {\left(q_2\right)}_{-}\ge {\left(q_1\right)}_{+}$ and $q_{*}^1={inf}_{l\in \mathbb{N}}{\left(q_2^1\right)}_l/{\left(q_1\right)}_{+}.$ Consequently, we get

$${\vartheta }_{12}\le C{\vartheta }_0:=C{∥{∥{\left\{{\hslash }_{ı}\right\}}_{ı}∥}_{{\ell }^r}∥}_{{\dot{K}}_{p(·)}^{\alpha ,q_1(·)}\left({\mathbb{R}}^n\right)}.$$

Now, let us turn to ${\vartheta }_{11}$. For any $\beta \in (0,1],(y,t)\in \mathsf{\Gamma }\left(x\right)$ and $\psi \in {\mathcal{C}}_{\beta }$, we have

$$\begin{array}{c}\hfill \underset{\psi \in {\mathcal{C}}_{\beta }}{sup}\left|\left({\hslash }_{ı}^j\right)\ast {\psi }_t\right|=\underset{\psi \in {\mathcal{C}}_{\beta }}{sup}\left|{\int }_{{\mathbb{R}}^n}{\psi }_t(y-z){\hslash }_{ı}^j\left(z\right)\right|\mathrm{d}z\lesssim \frac{1}{t^n}{\int }_{{\Re }_j\cap \{z:|y-z|\le t\}}\left|{\hslash }_{ı}^j\left(z\right)\right|\mathrm{d}z.\end{array}$$

Taking ${\parallel ·\parallel }_{{\ell }^r}$ of each side of the above inequality, we find that

$$\begin{array}{c}\hfill \parallel \{\underset{\psi \in {\mathcal{C}}_{\beta }}{sup}\left|\left({\hslash }_{ı}^j\right)\ast {\psi }_t\right|{\}}_{ı}{{\parallel }_{{\ell }^r}\lesssim \frac{1}{t^n}{\int }_{{\Re }_j\cap \{z:|y-z|\le t\}}\parallel \{{\hslash }_{ı}^j\left(z\right){\}}_{ı}\parallel }_{{\ell }^r}\mathrm{d}z.\end{array}$$

(10)

For any $x\in {\Re }_l,(y,t)\in \mathsf{\Gamma }\left(x\right),j\le l-2$ and $z\in {\Re }_j\cap \{z:|y-z|\le t\}$, it is easy to get

$$\begin{array}{c}\hfill t=(t+t)/2\ge \left(\right|x-y|+|y-z\left|\right)/2\ge |x-z|/2\ge \left|x\right|/4.\end{array}$$

(11)

Thus, by combining (10) and (11), together with Minkowski’s inequality, it follows that

$$\begin{array}{cc}\hfill {∥{\left\{S_{\beta }\left({\hslash }_{ı}^j\right)\left(x\right)\right\}}_{ı}∥}_{{\ell }^r}& \lesssim {\left({\int \int }_{\mathsf{\Gamma }\left(x\right)}{\left|\frac{1}{t^n}{\int }_{{\Re }_j\cap \{z:|y-z|\le t\}}\parallel \{{\hslash }_{ı}^j\left(z\right){\}}_{ı}{\parallel }_{{\ell }^r}\mathrm{d}z\right|}^2\frac{\mathrm{d}y\mathrm{d}t}{t^{1+n}}\right)}^{\frac{1}{2}}\hfill \\ & \lesssim {\left({\int }_{\left|x\right|/4}^{\infty }{\int }_{|x-y|<t}{\left|\frac{1}{t^n}{\int }_{{\Re }_j\cap \{z:|y-z|\le t\}}\parallel \{{\hslash }_{ı}^j\left(z\right){\}}_{ı}{\parallel }_{{\ell }^r}\mathrm{d}z\right|}^2\frac{\mathrm{d}y\mathrm{d}t}{t^{1+n}}\right)}^{\frac{1}{2}}\hfill \end{array}$$

$$\begin{array}{cc}& \lesssim \left({\int }_{{\Re }_j}\parallel \{{\hslash }_{ı}^j\left(z\right){\}}_{ı}{\parallel }_{{\ell }^r}\mathrm{d}z\right){\left({\int }_{\left|x\right|/4}^{\infty }\frac{\mathrm{d}t}{t^{1+2n}}\right)}^{\frac{1}{2}}\hfill \\ & \lesssim \left({\int }_{{\Re }_j}\parallel \{{\hslash }_{ı}^j\left(z\right){\}}_{ı}{\parallel }_{{\ell }^r}\mathrm{d}z\right){\left|x\right|}^n\lesssim 2^{-ln}{\int }_{{\Re }_j}\parallel \{{\hslash }_{ı}^j\left(z\right){\}}_{ı}{\parallel }_{{\ell }^r}\mathrm{d}z.\hfill \end{array}$$

Then, by virtue of Lemma 3, it follows that

$$\begin{array}{cc}& \sum _{l=-\infty }^{\infty }∥{\left(\frac{2^{l\alpha }\parallel {\sum }_{j=-\infty }^{l-3}\left\{S_{\beta }\left({\hslash }_{ı}^j\right)\right.{\}}_{ı}{\parallel }_{{\ell }^r}{\chi }_l}{{\vartheta }_0}\right)}^{q_2(·)}∥\hfill \\ \hfill \le & C\sum _{l=-\infty }^{\infty }{∥\frac{2^{l\alpha }\parallel {\sum }_{j=-\infty }^{l-3}\left\{S_{\beta }\left({\hslash }_{ı}^j\right)\right.{\}}_{ı}{\parallel }_{{\ell }^r}{\chi }_l}{{\vartheta }_0}∥}_{L^{\frac{p(·)}{q_2(·)}}}^{{\left(q_2^2\right)}_l}\hfill \\ \hfill \le & C\sum _{l=-\infty }^{\infty }{\left(\sum _{j=-\infty }^{l-3}{∥\frac{2^{l\alpha }\parallel \{S_{\beta }\left({\hslash }_{ı}^j\right){\}}_{ı}{\parallel }_{{\ell }^r}{\chi }_l}{{\vartheta }_0}∥}_{L^{p(·)}}\right)}^{{\left(q_2^2\right)}_l}\hfill \\ \hfill \le & C\sum _{l=-\infty }^{\infty }{\left(\sum _{j=-\infty }^{l-3}{∥\frac{2^{l\alpha }{2}^{-nl}{\int }_{{\Re }_j}{∥{\left\{{\hslash }_{ı}^j\right\}}_{ı}∥}_{{\ell }^r}\mathrm{d}z{\chi }_l}{{\vartheta }_0}∥}_{L^{p(·)}}\right)}^{{\left(q_2^2\right)}_l}\hfill \\ \hfill \le & C\sum _{l=-\infty }^{\infty }{\left(\sum _{j=-\infty }^{l-3}{∥{∥{\left\{\frac{{\hslash }_{ı}^j}{{\vartheta }_0}\right\}}_{ı}∥}_{{\ell }^r}∥}_{L^1\left({\mathbb{R}}^n\right)}{2}^{l\alpha }{2}^{-nl}{∥{\chi }_l∥}_{L^{p(·)}}\right)}^{{\left(q_2^2\right)}_l}\hfill \end{array}$$

where

$${\left(q_2^2\right)}_l=\left\{\begin{array}{c}{\left(q_2\right)}_{-}\mathrm{if}{∥\frac{2^{l\alpha }∥{\sum }_{j=-\infty }^{l-3}{\left\{S_{\beta }\left({\hslash }_{ı}^j\right)\right\}}_{ı}{\parallel }_{{\ell }^r}{\chi }_l}{{\vartheta }_0}∥}_{L^{p(·)}}\le 1,\hfill \\ {\left(q_2\right)}_{+}\mathrm{otherwise}.\hfill \end{array}\right.$$

From Hölder’s inequality (4), and using Lemmas 1 and 2, we infer that

$$\begin{array}{cc}& \sum _{l=-\infty }^{\infty }∥{\left(\frac{2^{l\alpha }\parallel {\sum }_{j=-\infty }^{l-3}\left\{S_{\beta }\left({\hslash }_{ı}^j\right)\right.{\}}_{ı}{\parallel }_{{\ell }^r}{\chi }_l}{{\vartheta }_0}\right)}^{q_2(·)}∥\hfill \\ & \le C\sum _{l=-\infty }^{\infty }{\left(\sum _{j=-\infty }^{l-3}{∥{∥{\left\{\frac{{\hslash }_{ı}^j}{{\vartheta }_0}\right\}}_{ı}∥}_{{\ell }^r}∥}_{L^{p(·)}}{2}^{l\alpha }\frac{{∥{\chi }_j∥}_{L^{p^{\prime }(·)}}}{{∥{\chi }_l∥}_{L^{p^{\prime }(·)}}}\frac{|B_l|}{2^{nl}}\right)}^{{\left(q_2^2\right)}_l}\hfill \\ & \le C\sum _{l=-\infty }^{\infty }{\left\{\sum _{j=-\infty }^{l-3}{∥{\left(\frac{2^{j\alpha }∥{\left\{{\hslash }_{ı}^{}\right\}}_{ı}{\parallel }_{{\ell }^r}{\chi }_j}{{\vartheta }_0}\right)}^{q_1(·)}∥}_{L^{\frac{p(·)}{q_1(·)}}}^{\frac{1}{{\left(q_1\right)}_{+}}}{2}^{(l-j)(\alpha -{\delta }_{12})}\right\}}^{{\left(q_2^2\right)}_l}.\hfill \end{array}$$

Now, we can distinguish the following two cases: ${\left(q_1\right)}_{+}>1$ and $0<{\left(q_1\right)}_{+}\le 1$.

Case$1^{\circ }$: If $0<{\left(q_1\right)}_{+}\le 1$, then from Proposition 1 and Lemma 3, we obtain

$$\begin{array}{cc}& \sum _{l=-\infty }^{\infty }∥{\left(\frac{2^{l\alpha }\parallel {\sum }_{j=-\infty }^{l-3}\left\{S_{\beta }\left({\hslash }_{ı}^j\right)\right.{\}}_{ı}{\parallel }_{{\ell }^r}{\chi }_l}{{\vartheta }_0}\right)}^{q_2(·)}∥\hfill \\ \hfill \le & C\sum _{l=-\infty }^{\infty }{\left\{\sum _{j=-\infty }^{l-3}{∥{\left(\frac{2^{j\alpha }∥{\left\{{\hslash }_{ı}^{}\right\}}_{ı}{\parallel }_{{\ell }^r}{\chi }_j}{{\vartheta }_0}\right)}^{q_1(·)}∥}_{L^{\frac{p(·)}{q_1(·)}}}{2}^{{\left(q_1\right)}_{+}(l-j)(\alpha -{\delta }_{12})}\right\}}^{\frac{{\left(q_2^2\right)}_l}{{\left(q_1\right)}_{+}}}\hfill \end{array}$$

$$\begin{array}{cc}\hfill \le & C{\left\{\sum _{j=-\infty }^{\infty }{∥{\left(\frac{2^{j\alpha }∥{\left\{{\hslash }_{ı}^{}\right\}}_{ı}{\parallel }_{{\ell }^r}{\chi }_j}{{\vartheta }_0}\right)}^{q_1(·)}∥}_{L^{\frac{p(·)}{q_1(·)}}}\sum _{l=j+3}^{\infty }{2}^{{\left(q_1\right)}_{+}(l-j)(\alpha -{\delta }_{12})}\right\}}^{q_{*}^2}\le C,\hfill \end{array}$$

where $q_{*}^2={inf}_{l\in \mathbb{N}}{\left(q_2^2\right)}_l/{\left(q_1\right)}_{+}$.

Case$2^{\circ }$: If ${\left(q_1\right)}_{+}>1$, then $1\le {\left(q_1\right)}_{+}\le {\left(q_2\right)}_{-}\le {\left(q_2^2\right)}_l$. Hence, for $\alpha <n{\delta }_{12}$, by using Hölder’s inequality (4) and Proposition 1, we obtain that

$$\begin{array}{cc}& \sum _{l=-\infty }^{\infty }∥{\left(\frac{2^{l\alpha }\parallel {\sum }_{j=-\infty }^{l-3}\left\{S_{\beta }\left({\hslash }_{ı}^j\right)\right.{\}}_{ı}{\parallel }_{{\ell }^r}{\chi }_l}{{\vartheta }_0}\right)}^{q_2(·)}∥\hfill \\ & \le C\sum _{l=-\infty }^{\infty }{\left\{\sum _{j=-\infty }^{l-3}{∥{\left(\frac{2^{j\alpha }∥{\left\{{\hslash }_{ı}^{}\right\}}_{ı}{\parallel }_{{\ell }^r}{\chi }_j}{{\vartheta }_0}\right)}^{q_1(·)}∥}_{L^{\frac{p(·)}{q_1(·)}}}{2}^{\frac{1}{2}(l-j)(\alpha -{\delta }_{12}){\left(q_1\right)}_{+}}\right\}}^{\frac{{\left(q_2^2\right)}_l}{{\left(q_1\right)}_{+}}}\hfill \\ & ·{\left\{\sum _{j=-\infty }^{l-3}{2}^{\frac{1}{2}(l-j)(\alpha -{\delta }_{12})b}\right\}}^{\frac{{\left(q_2^2\right)}_l}{b}}\hfill \\ & \le C{\left\{\sum _{j=-\infty }^{\infty }{∥{\left(\frac{2^{j\alpha }∥{\left\{{\hslash }_{ı}^{}\right\}}_{ı}{\parallel }_{{\ell }^r}{\chi }_j}{{\vartheta }_0}\right)}^{q_1(·)}∥}_{L^{\frac{p(·)}{q_1(·)}}}\sum _{l=j+3}^{\infty }{2}^{\frac{1}{2}(l-j)(\alpha -{\delta }_{12}){\left(q_1\right)}_{+}}\right\}}^{q_{*}^2}\le C,\hfill \end{array}$$

where $b={\left({\left(q_1\right)}_{+}\right)}^{\prime }$ and $q_{*}^2={inf}_{l\in \mathbb{N}}{\left(q_2^2\right)}_l/{\left(q_1\right)}_{+}$.

Then, based on the computations presented above, it follows that

$${\vartheta }_{11}\le C{\vartheta }_0:=C{∥{∥{\left\{{\hslash }_{ı}\right\}}_{ı}∥}_{{\ell }^r}∥}_{{\dot{K}}_{p(·)}^{\alpha ,q_1(·)}\left({\mathbb{R}}^n\right)}.$$

Finally, we estimate ${\vartheta }_{13}$. For any $\beta \in (0,1],(y,t)\in \mathsf{\Gamma }\left(x\right)$ and $\psi \in {\mathcal{C}}_{\beta }$, we have

$$\begin{array}{c}\hfill \underset{\psi \in {\mathcal{C}}_{\beta }}{sup}\left|\left({\hslash }_{ı}^j\right)\ast {\psi }_t\right|=\underset{\psi \in {\mathcal{C}}_{\beta }}{sup}\left|{\int }_{{\mathbb{R}}^n}{\psi }_t(y-z){\hslash }_{ı}^j\left(z\right)\right|\mathrm{d}z\lesssim \frac{1}{t^n}{\int }_{{\Re }_j\cap \{z:|y-z|\le t\}}\left|{\hslash }_{ı}^j\left(z\right)\right|\mathrm{d}z.\end{array}$$

By applying the norm ${\parallel ·\parallel }_{{\ell }^r}$ on each side of the above inequality, we find that

$$\begin{array}{c}\hfill \parallel \{\underset{\psi \in {\mathcal{C}}_{\beta }}{sup}\left|\left({\hslash }_{ı}^j\right)\ast {\psi }_t\right|{\}}_{ı}{{\parallel }_{{\ell }^r}\lesssim \frac{1}{t^n}{\int }_{{\Re }_j\cap \{z:|y-z|\le t\}}\parallel \{{\hslash }_{ı}^j\left(z\right){\}}_{ı}\parallel }_{{\ell }^r}\mathrm{d}z.\end{array}$$

(12)

Let $z\in {\Re }_j\cap \{z:|y-z|\le t\},(y,t)\in \mathsf{\Gamma }\left(x\right)$ and $j\ge l+3$, it is easy to get

$$\begin{array}{c}\hfill t=(t+t)/2\ge \left(\right|x-y|+|y-z\left|\right)/2\ge |x-z|/2\ge \left|\right|z|-|x\left|\right|/2\ge \left|z\right|/4.\end{array}$$

(13)

Thus, by combining (12) and (13), we obtain

$$\begin{array}{cc}\hfill {∥{\left\{S_{\beta }\left({\hslash }_{ı}^j\right)\left(x\right)\right\}}_{ı}∥}_{{\ell }^r}& \lesssim {\left({\int \int }_{\mathsf{\Gamma }\left(x\right)}{\left|\frac{1}{t^n}{\int }_{{\Re }_j\cap \{z:|y-z|\le t\}}{∥{\left\{{\hslash }_{ı}^j\left(z\right)\right\}}_{ı}∥}_{{\ell }^r}\mathrm{d}z\right|}^2\frac{\mathrm{d}y\mathrm{d}t}{t^{1+n}}\right)}^{\frac{1}{2}}\hfill \\ & \lesssim {\left({\int }_{\left|z\right|/4}^{\infty }{\int }_{|x-y|<t}{\left|\frac{1}{t^n}{\int }_{{\Re }_j\cap \{z:|y-z|\le t\}}{∥{\left\{{\hslash }_{ı}^j\left(z\right)\right\}}_{ı}∥}_{{\ell }^r}\mathrm{d}z\right|}^2\frac{\mathrm{d}y\mathrm{d}t}{t^{1+n}}\right)}^{\frac{1}{2}}\hfill \\ & \lesssim \left({\int }_{{\Re }_j}{∥{\left\{{\hslash }_{ı}^j\left(z\right)\right\}}_{ı}∥}_{{\ell }^r}\mathrm{d}z\right){\left({\int }_{\left|z\right|/4}^{\infty }\frac{dt}{t^{1+2n}}\right)}^{\frac{1}{2}}\hfill \\ & \lesssim \left({\int }_{{\Re }_j}{∥{\left\{{\hslash }_{ı}^j\left(z\right)\right\}}_{ı}∥}_{{\ell }^r}\mathrm{d}z\right){\left|z\right|}^n\lesssim 2^{-nj}{\int }_{{\Re }_j}{∥{\left\{{\hslash }_{ı}^j\left(z\right)\right\}}_{ı}∥}_{{\ell }^r}\mathrm{d}z.\hfill \end{array}$$

Then, from Lemmas 1–3, it follows that

$$\begin{array}{cc}& \sum _{l=-\infty }^{\infty }∥{\left(\frac{2^{l\alpha }\parallel {\sum }_{j=l+3}^{\infty }\left\{S_{\beta }\left({\hslash }_{ı}^j\right)\right.{\}}_{ı}{\parallel }_{{\ell }^r}{\chi }_l}{{\vartheta }_0}\right)}^{q_2(·)}∥\hfill \\ \hfill \le & C\sum _{l=-\infty }^{\infty }{∥\frac{2^{l\alpha }\parallel {\sum }_{j=l+3}^{\infty }\left\{S_{\beta }\left({\hslash }_{ı}^j\right)\right.{\}}_{ı}{\parallel }_{{\ell }^r}{\chi }_l}{{\vartheta }_0}∥}_{L^{\frac{p(·)}{q_2(·)}}}^{{\left(q_2^3\right)}_l}\hfill \\ \hfill \le & C\sum _{l=-\infty }^{\infty }{\left\{\sum _{j=l+3}^{\infty }{∥\frac{2^{l\alpha }{2}^{-nj}{\int }_{{\Re }_j}{∥{\left\{{\hslash }_{ı}^j\right\}}_{ı}∥}_{{\ell }^r}\mathrm{d}z{\chi }_l}{{\vartheta }_0}∥}_{L^{p(·)}}\right\}}^{{\left(q_2^3\right)}_l}\hfill \\ \hfill \le & C\sum _{l=-\infty }^{\infty }{\left\{\sum _{j=l+3}^{\infty }{∥{∥{\left\{\frac{{\hslash }_{ı}^j}{{\vartheta }_0}\right\}}_{ı}∥}_{{\ell }^r}∥}_{L^1\left({\mathbb{R}}^n\right)}{2}^{l\alpha }{2}^{-nj}{∥{\chi }_l∥}_{L^{p(·)}}\right\}}^{{\left(q_2^3\right)}_l}\hfill \\ \hfill \le & C\sum _{l=-\infty }^{\infty }{\left\{\sum _{j=l+3}^{\infty }{∥{∥{\left\{\frac{{\hslash }_{ı}^j}{{\vartheta }_0}\right\}}_{ı}∥}_{{\ell }^r}∥}_{L^{p(·)}\left({\mathbb{R}}^n\right)}{2}^{l\alpha }\frac{{∥{\chi }_l∥}_{L^{p(·)}}}{{∥{\chi }_j∥}_{L^{p(·)}}}\frac{|B_j|}{2^{nj}}\right\}}^{{\left(q_2^3\right)}_l}\hfill \\ \hfill \le & C\sum _{l=-\infty }^{\infty }{\left\{\sum _{j=l+3}^{\infty }{∥{\left(\frac{2^{j\alpha }∥{\left\{{\hslash }_{ı}^{}\right\}}_{ı}{\parallel }_{{\ell }^r}{\chi }_j}{{\vartheta }_0}\right)}^{q_1(·)}∥}_{L^{\frac{p(·)}{q_1(·)}}}^{\frac{1}{{\left(q_1\right)}_{+}}}{2}^{(l-j)(\alpha +{\delta }_{11})}\right\}}^{{\left(q_2^3\right)}_l}.\hfill \end{array}$$

where

$${\left(q_2^3\right)}_l=\left\{\begin{array}{c}{\left(q_2\right)}_{-}\mathrm{if}{∥\frac{2^{l\alpha }∥{\sum }_{j=l+3}^{\infty }{\left\{S_{\beta }\left({\hslash }_{ı}^j\right)\right\}}_{ı}{\parallel }_{{\ell }^r}{\chi }_l}{{\vartheta }_0}∥}_{L^{p(·)}}\le 1,\hfill \\ {\left(q_2\right)}_{+}\mathrm{otherwise}.\hfill \end{array}\right.$$

Observing that ${\left(q_1\right)}_{+}\le {\left(q_2\right)}_{-}$ and the fact $\alpha >-n{\delta }_{11}$, by using a similar argument to that used for ${\vartheta }_{11}$, we deduce that

$${\vartheta }_{13}\le C{\vartheta }_0:=C{∥{∥{\left\{{\hslash }_{ı}\right\}}_{ı}∥}_{{\ell }^r}∥}_{{\dot{K}}_{p(·)}^{\alpha ,q_1(·)}\left({\mathbb{R}}^n\right)}.$$

This completes the proof of Theorem 1. □

3.2. Intrinsic Square Function

Theorem 2.

Suppose that $q_1(·)\mathrm{and}{q}_2(·)\in {\mathcal{P}}_0^{*}\left({\mathbb{R}}^n\right)$ with ${\left(q_1\right)}_{+}\le {\left(q_2\right)}_{-}$ and let $\beta \in (0,1],p(·)\in {\mathcal{B}}^{*}\left({\mathbb{R}}^n\right)$. If $\alpha \in (-n{\delta }_{11},n{\delta }_{12})$, where ${\delta }_{11},{\delta }_{12}\in (0,1)$ are constants stated in Lemma 2, then $S_{\beta }$ is bounded from ${\dot{K}}_{p(·)}^{\alpha ,q_1(·)}\left({\mathbb{R}}^n\right)$ to ${\dot{K}}_{p(·)}^{\alpha ,q_2(·)}\left({\mathbb{R}}^n\right)$.

Proof. 

Let $\hslash \in {\dot{K}}_{p(·)}^{\alpha ,q_1(·)}\left({\mathbb{R}}^n\right)$. We can write $\hslash \left(x\right)={\sum }_{j=-\infty }^{\infty }\hslash \left(x\right){\chi }_j\left(x\right)={\sum }_{j=-\infty }^{\infty }{\hslash }_j\left(x\right).$ By the definition of ${\dot{K}}_{p(·)}^{\alpha ,q(·)}\left({\mathbb{R}}^n\right)$, we have

$${∥S_{\beta }(\hslash )∥}_{{\dot{K}}_{p(·)}^{\alpha ,q_2(·)}\left({\mathbb{R}}^n\right)}=inf\left\{\vartheta >0:\sum _{l=-\infty }^{\infty }{∥{\left(\frac{2^{l\alpha }\left|S_{\beta }(\hslash ){\chi }_l\right|}{\vartheta }\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}\le 1\right\}.$$

Since

$$\begin{array}{cc}\hfill {∥{\left(\frac{2^{l\alpha }\left|S_{\beta }(\hslash ){\chi }_l\right|}{\vartheta }\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}& \le {∥{\left(\frac{2^{l\alpha }\left|{\sum }_{j=-\infty }^{\infty }{S}_{\beta }\left({\hslash }_j\right){\chi }_l\right|}{{\vartheta }_{21}+{\vartheta }_{22}+{\vartheta }_{23}}\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}\hfill \\ & \le C{∥{\left(\frac{2^{l\alpha }\left|{\sum }_{j=-\infty }^{l-3}{S}_{\beta }\left({\hslash }_j\right){\chi }_l\right|}{{\vartheta }_{21}}\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}\hfill \\ & +C{∥{\left(\frac{2^{l\alpha }\left|{\sum }_{j=l-2}^{l+2}{S}_{\beta }\left({\hslash }_j\right){\chi }_l\right|}{{\vartheta }_{22}}\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}\hfill \\ & +C{∥{\left(\frac{2^{l\alpha }\left|{\sum }_{j=l+2}^{\infty }{S}_{\beta }\left({\hslash }_j\right){\chi }_l\right|}{{\vartheta }_{23}}\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}},\hfill \end{array}$$

here

$$\begin{array}{cc}& {\vartheta }_{21}={∥{\left\{2^{l\alpha }\left|\sum _{j=-\infty }^{l-3}{S}_{\beta }\left({\hslash }_j\right){\chi }_l\right|\right\}}_{l=-\infty }∥}_{{\ell }^{q_2(·)}\left(L^{p(·)}\right)},\hfill \\ & {\vartheta }_{22}={∥{\left\{2^{l\alpha }\left|\sum _{j=l-2}^{l+2}{S}_{\beta }\left({\hslash }_j\right){\chi }_l\right|\right\}}_{l=-\infty }^{\infty }∥}_{{\ell }^{q_2(·)}\left(L^{p(·)}\right)},\hfill \\ & {\vartheta }_{23}={∥{\left\{2^{l\alpha }\left|\sum _{j=l+3}^{\infty }{S}_{\beta }\left({\hslash }_j\right){\chi }_l\right|\right\}}_{l=-\infty }^{\infty }∥}_{{\ell }^{q_2(·)}\left(L^{p(·)}\right)},\hfill \end{array}$$

and $\vartheta :=({\vartheta }_{21}+{\vartheta }_{22}+{\vartheta }_{23}):={\sum }_{\varrho =1}^3{\vartheta }_{2\varrho }.$ That is,

$${∥S_{\beta }(\hslash )∥}_{{\dot{K}}_{p(·)}^{\alpha ,q_2(·)}\left({\mathbb{R}}^n\right)}\le C\vartheta :=C\left(\sum _{\varrho =1}^3{\vartheta }_{1\varrho }\right).$$

Hence, once we establish that

$${\vartheta }_{21}\le C{∥\hslash ∥}_{{\dot{K}}_{p(·)}^{\alpha ,q_1(·)}\left({\mathbb{R}}^n\right)},{\vartheta }_{22}\le C{∥\hslash ∥}_{{\dot{K}}_{p(·)}^{\alpha ,q_1(·)}\left({\mathbb{R}}^n\right)},{\vartheta }_{23}\le C{∥\hslash ∥}_{{\dot{K}}_{p(·)}^{\alpha ,q_1(·)}\left({\mathbb{R}}^n\right)},$$

we are done. Let ${\vartheta }_{20}={\parallel \hslash \parallel }_{{\dot{K}}_{p(·)}^{\alpha ,q_1(·)}\left({\mathbb{R}}^n\right)}$.

First, we consider ${\vartheta }_{22}.$ By the boundedness of $S_{\beta }$ on $L^{p(·)}$, and using the same technique as for ${\vartheta }_{12}$ in the proof of Theorem 1, we find immediately that

$$\sum _{l=-\infty }^{\infty }{∥{\left(\frac{2^{l\alpha }\left|{\sum }_{j=l-2}^{l+2}{S}_{\beta }\left({\hslash }_j\right){\chi }_l\right|}{{\vartheta }_{20}}\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}\le C.$$

That is,

$${\vartheta }_{22}\le C{\vartheta }_{20}:=C{\parallel \hslash \parallel }_{{\dot{K}}_{p(·)}^{\alpha ,q_1(·)}\left({\mathbb{R}}^n\right)}.$$

Now, let us turn to the estimates of ${\vartheta }_{21}$. For any $\beta \in (0,1],x\in {\Re }_l,(y,t)\in \mathsf{\Gamma }\left(x\right)$ and $j\le l-3$, we have

$$\begin{array}{c}\hfill \underset{\psi \in {\mathcal{C}}_{\beta }}{sup}\left|\left({\hslash }_j\right)\ast {\psi }_t\right|=\underset{\psi \in {\mathcal{C}}_{\beta }}{sup}\left|{\int }_{{\mathbb{R}}^n}{\psi }_t(y-z){\hslash }_j\left(z\right)\mathrm{d}z\right|\lesssim \frac{1}{t^n}{\int }_{{\Re }_j\cap \{z:|y-z|\le t\}}\left|{\hslash }_j\left(z\right)\mathrm{d}z\right|.\end{array}$$

(14)

Let $z\in {\Re }_j\cap \{z:|y-z|\le t\}$. Since $(y,t)\in \mathsf{\Gamma }\left(x\right)$, we have

$$\begin{array}{c}\hfill t=(t+t)/2\ge \left(\right|x-y|+|y-z\left|\right)/2\ge |x-z|/2\ge \left|x\right|/4.\end{array}$$

(15)

Thus, by combining (14) and (15), we deduce that

$$\begin{array}{cc}\hfill S_{\beta }\left({\hslash }_j\right)\left(x\right)& \lesssim {\left({\int \int }_{\mathsf{\Gamma }\left(x\right)}{\left|\frac{1}{t^n}{\int }_{{\Re }_j\cap \{z:|y-z|\le t\}}\left|{\hslash }_j\left(z\right)\right|\mathrm{d}z\right|}^2\frac{\mathrm{d}y\mathrm{d}t}{t^{1+n}}\right)}^{\frac{1}{2}}\hfill \\ & \lesssim {\left({\int }_{\left|x\right|/4}^{\infty }{\int }_{|x-y|<t}{\left|\frac{1}{t^n}{\int }_{{\Re }_j\cap \{z:|y-z|\le t\}}\left|{\hslash }_j\left(z\right)\right|\mathrm{d}z\right|}^2\frac{\mathrm{d}y\mathrm{d}t}{t^{1+n}}\right)}^{\frac{1}{2}}\hfill \\ & \lesssim \left({\int }_{{\Re }_j}\left|{\hslash }_j\left(z\right)\right|\mathrm{d}z\right){\left({\int }_{\left|x\right|/4}^{\infty }\frac{\mathrm{d}t}{t^{1+2n}}\right)}^{\frac{1}{2}}\hfill \\ & \lesssim \left({\int }_{{\Re }_j}\left|{\hslash }_j\left(z\right)\right|\mathrm{d}z\right){\left|x\right|}^n\lesssim 2^{-ln}{∥{\hslash }_j∥}_{L^1\left({\mathbb{R}}^n\right)}.\hfill \end{array}$$

Then, by applying Lemma 3 and Hölder’s inequality (4), we find that

$$\begin{array}{cc}& \sum _{l=-\infty }^{\infty }{∥{\left(\frac{2^{l\alpha }\left|{\sum }_{j=-\infty }^{l-3}{S}_{\beta }\left({\hslash }_j\right){\chi }_l\right|}{{\vartheta }_{20}}\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}\hfill \\ & \le C\sum _{l=-\infty }^{\infty }{∥\frac{\left.2^{l\alpha }\left|{\sum }_{j=-\infty }^{l-3}\right.S_{\beta }\left({\hslash }_j\right){\chi }_l\right|}{{\vartheta }_{20}}∥}_{L^{p(·)}}^{{\left(q_2^1\right)}_l}\hfill \\ & \le C\sum _{l=-\infty }^{\infty }{\left(2^{l\alpha }\sum _{j=-\infty }^{l-3}{∥\frac{\left|{\hslash }_j\right|}{{\vartheta }_{20}}∥}_{L^{p(·)}}{∥{\chi }_l∥}_{L^{p(·)}}{∥{\chi }_j∥}_{L^{p_1^{\prime }(·)}}{2}^{-nl}\right)}^{{\left(q_2^1\right)}_l}\hfill \end{array}$$

where

$${\left(q_2^1\right)}_l=\left\{\begin{array}{c}{\left(q_2\right)}_{-}\mathrm{if}{∥\frac{2^{l\alpha }\left|{\sum }_{j=-\infty }^{l-3}{S}_{\beta }\left({\hslash }_j\right)\right|{\chi }_l}{{\vartheta }_{20}}∥}_{L^{p(·)}}\le 1,\hfill \\ {\left(q_2\right)}_{+}\mathrm{otherwise}.\hfill \end{array}\right.$$

From Lemmas 1–3, we deduce that

$$\begin{array}{cc}& \sum _{l=-\infty }^{\infty }{∥{\left(\frac{2^{l\alpha }\left|{\sum }_{j=-\infty }^{l-3}{S}_{\beta }\left({\hslash }_j\right){\chi }_l\right|}{{\vartheta }_{20}}\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}\hfill \\ & \le C\sum _{l=-\infty }^{\infty }{\left(2^{l\alpha }\sum _{j=-\infty }^{l-3}{∥\frac{|{\hslash }_j|}{{\vartheta }_{20}}∥}_{L^{p(·)}}\frac{{∥{\chi }_j∥}_{L^{p_1^{\prime }(·)}}}{{∥{\chi }_l∥}_{L^{p_1^{\prime }(·)}}}\frac{|B_l|}{2^{nl}}\right)}^{{\left(q_2^1\right)}_l}\hfill \\ & \le C\sum _{l=-\infty }^{\infty }{\left\{\sum _{j=-\infty }^{l-3}{∥{\left(\frac{\left|2^{j\alpha }\hslash {\chi }_j\right|}{{\vartheta }_{20}}\right)}^{q_1(·)}∥}_{L^{\frac{p(·)}{q_1(·)}}}^{\frac{1}{{\left(q_1\right)}_{+}}}{2}^{(l-j)\left(\alpha -n{\delta }_{12}\right)}\right\}}^{{\left(q_2^1\right)}_l}\hfill \end{array}$$

Moreover, using the same approach as ${\vartheta }_{11}$ in the proof of Theorem 1, we infer that

$$\begin{array}{cc}& \sum _{l=-\infty }^{\infty }{∥{\left(\frac{2^{l\alpha }\left|{\sum }_{j=-\infty }^{l-3}{S}_{\beta }\left({\hslash }_j\right){\chi }_l\right|}{{\vartheta }_{20}}\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}\hfill \\ & \le C\left\{\begin{array}{c}{\left\{{\displaystyle \sum _{j=-\infty }^{\infty }}{∥{\left(\frac{2^{j\alpha }\left|\hslash {\chi }_j\right|}{{\vartheta }_{20}}\right)}^{q_1(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}{\displaystyle \sum _{l=j+3}^{\infty }}{2}^{\frac{1}{2}(l-j)\left(\alpha -n{\delta }_{12}\right){\left(q_1\right)}_{+}}\right\}}^{q_{*}^1},{\left(q_1\right)}_{+}>1,\hfill \\ \\ {\left\{{\displaystyle \sum _{j=-\infty }^{\infty }}{∥{\left(\frac{2^{j\alpha }\left|\hslash {\chi }_j\right|}{{\vartheta }_{20}}\right)}^{q_1(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}{\displaystyle \sum _{l=j+3}^{\infty }}{2}^{(l-j)\left(\alpha -n{\delta }_{12}\right){\left(q_1\right)}_{+}}\right\}}^{q_{*}^1},0<{\left(q_1\right)}_{+}\le 1,\hfill \end{array}\right.\hfill \end{array}$$

where $q_{*}^1={inf}_{l\in \mathbb{N}}{\left(q_2^1\right)}_l/{\left(q_1\right)}_{+}$. This implies that

$${\vartheta }_{21}\le C{\vartheta }_{20}:=C{\parallel \hslash \parallel }_{{\dot{K}}_p^{\alpha ,q_1(·)}\left({\mathbb{R}}^n\right)}.$$

Finally, we estimate ${\vartheta }_{23}.$ For any $x\in {\Re }_j,(y,t)\in \mathsf{\Gamma }\left(x\right),z\in {\Re }_j\cap \{z:|y-z|\le t\}$ and $j\ge l+3$, we find that

$$\begin{array}{c}\hfill t=(t+t)/2\ge \left(\right|x-y|+|y-z\left|\right)/2\ge |x-z|/2\ge \left|\right|z|-|x\left|\right|/2\ge \left|z\right|/4,\end{array}$$

(16)

Thus, by combining (14) and (16), we get

$$\begin{array}{cc}\hfill S_{\beta }\left({\hslash }_j\right)\left(x\right)& \lesssim {\left({\int \int }_{\mathsf{\Gamma }\left(x\right)}{\left|\frac{1}{t^n}{\int }_{{\Re }_j\cap \{z:|y-z|\le t\}}\left|{\hslash }_j\left(z\right)\right|\mathrm{d}z\right|}^2\frac{\mathrm{d}y\mathrm{d}t}{t^{1+n}}\right)}^{\frac{1}{2}}\hfill \\ & \lesssim {\left({\int }_{\left|z\right|/4}^{\infty }{\int }_{|x-y|<t}{\left|\frac{1}{t^n}{\int }_{{\Re }_j\cap \{z:|y-z|\le t\}}\left|{\hslash }_j\left(z\right)\right|\mathrm{d}z\right|}^2\frac{\mathrm{d}y\mathrm{d}t}{t^{1+n}}\right)}^{\frac{1}{2}}\hfill \\ & \lesssim \left({\int }_{{\Re }_j}\left|{\hslash }_j\left(z\right)\right|\mathrm{d}z\right){\left({\int }_{\left|z\right|/4}^{\infty }\frac{\mathrm{d}t}{t^{1+2n}}\right)}^{\frac{1}{2}}\hfill \\ & \lesssim \left({\int }_{{\Re }_j}\left|{\hslash }_j\left(z\right)\right|\mathrm{d}z\right){\left|z\right|}^n\lesssim 2^{-jn}{∥{\hslash }_j∥}_{L^1\left({\mathbb{R}}^n\right)}.\hfill \end{array}$$

Thus, when $\alpha >-n{\delta }_{11}$, as argued for ${\vartheta }_{21}$, we deduce that

$$\begin{array}{cc}& \sum _{l=-\infty }^{\infty }{∥{\left(\frac{2^{l\alpha }\left|{\sum }_{j=l+3}^{\infty }{S}_{\beta }\left({\hslash }_j\right){\chi }_l\right|}{{\vartheta }_{20}}\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}\hfill \\ & \le C\sum _{l=-\infty }^{\infty }{∥\frac{\left.2^{l\alpha }\left|{\sum }_{j=l+3}^{\infty }\right.S_{\beta }\left({\hslash }_j\right){\chi }_l\right|}{{\vartheta }_{20}}∥}_{L^{p(·)}}^{{\left(q_2^2\right)}_l}\hfill \\ & \le C\sum _{l=-\infty }^{\infty }{\left\{2^{l\alpha }\sum _{j=l+3}^{\infty }{∥\frac{|{\hslash }_j|}{{\vartheta }_{20}}∥}_{L^{p(·)}}{∥{\chi }_l∥}_{L^{p(·)}}{∥{\chi }_j∥}_{L^{p_1^{\prime }(·)}}{2}^{-nj}\right\}}^{{\left(q_2^2\right)}_l}\hfill \\ & \le C\sum _{l=-\infty }^{\infty }{\left\{2^{l\alpha }\sum _{j=l+3}^{\infty }{2}^{-j\alpha }{∥\frac{2^{j\alpha }\left|{\hslash }_j\right|}{{\vartheta }_{20}}∥}_{L^{p(·)}}\frac{{∥{\chi }_l∥}_{L^{p(·)}}}{{∥{\chi }_j∥}_{L^{p(·)}}}\frac{|B_j|}{2^{nj}}\right\}}^{{\left(q_2^2\right)}_l}\hfill \\ & \le C\sum _{l=-\infty }^{\infty }{\left\{\sum _{j=l+3}^{\infty }{∥{\left(\frac{2^{j\alpha }\left|\hslash {\chi }_j\right|}{{\vartheta }_{20}}\right)}^{q_1(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}^{\frac{1}{{\left(q_1\right)}_{+}}}{2}^{(l-j)(\alpha +n{\delta }_{11})}\right\}}^{{\left(q_2^2\right)}_l}\le C,\hfill \end{array}$$

here

$${\left(q_2^2\right)}_l=\left\{\begin{array}{c}{\left(q_2\right)}_{-}\mathrm{if}{∥\frac{2^{l\alpha }\left|{\sum }_{j=l+3}^{\infty }{S}_{\beta }\left({\hslash }_j\right){\chi }_l\right|}{{\vartheta }_{20}}∥}_{L^{p(·)}}\le 1,\hfill \\ {\left(q_2\right)}_{+}\mathrm{otherwise}.\hfill \end{array}\right.$$

Hence,

$${\vartheta }_{23}\le C{\vartheta }_{20}:=C{\parallel \hslash \parallel }_{{\dot{K}}_{p(·)}^{\alpha ,q_1(·)}\left({\mathbb{R}}^n\right)}.$$

This finishes the proof of Theorem 2. □

3.3. Commutator of the Intrinsic Square Function

Let B be any ball with a radius of $r>0$ and a centre at $x\in {\mathbb{R}}^n$. If a locally integrable function b satisfies the following conditions, it is said to be a $\mathrm{BMO}$ function.

$${\parallel b\parallel }_{*}:=\underset{x\in {\mathbb{R}}^n,r>0}{sup}{\left|B\right|}^{-1}{\int }_B\left|b\left(y\right)-b_B\right|\mathrm{d}y<\infty ,$$

where ${\parallel b\parallel }_{*}$ is the norm in $BMO\left({\mathbb{R}}^n\right)$ and $b_B:={\left|B\right|}^{-1}{\int }_{B}b\left(t\right)\mathrm{d}t$.

The commutator of the intrinsic square function $S_{\beta }(\hslash )$ and $b\in BMO\left({\mathbb{R}}^n\right)$ is defined as follows:

$$\left[b,S_{\beta }\right](\hslash )\left(x\right)={\left({\int \int }_{\mathsf{\Gamma }\left(x\right)}\underset{\psi \in {\mathcal{C}}_{\beta }}{sup}{\left|{\int }_{{\mathbb{R}}^n}(b\left(x\right)-b\left(z\right)){\psi }_t(y-z)\hslash \left(z\right)\mathrm{d}z\right|}^2\frac{dydt}{t^{1+n}}\right)}^{\frac{1}{2}}.$$

Theorem 3.

Suppose that $q_1(·)\mathrm{and}{q}_2(·)\in {\mathcal{P}}_0^{*}\left({\mathbb{R}}^n\right)$ with ${\left(q_1\right)}_{+}\le {\left(q_2\right)}_{-}$ and let $\beta \in (0,1],p(·)\in {\mathcal{B}}^{*}\left({\mathbb{R}}^n\right),b\in BMO\left({\mathbb{R}}^n\right)$. If $\alpha \in (-n{\delta }_{11},n{\delta }_{12})$, where ${\delta }_{11},{\delta }_{12}\in (0,1)$ are constants stated in Lemma 2, then $[b,S_{\beta }]$ is bounded from ${\dot{K}}_{p(·)}^{\alpha ,q_1(·)}\left({\mathbb{R}}^n\right)$ to ${\dot{K}}_{p(·)}^{\alpha ,q_2(·)}\left({\mathbb{R}}^n\right)$.

To give the proof Theorem 3, we use the following lemmas.

Lemma 6

([25]). Let $b\in BMO\left({\mathbb{R}}^n\right),\upsilon$ be a positive integer and $p(·)\in {\mathcal{B}}^{*}\left({\mathbb{R}}^n\right)$. Then there is a positive C such that for each $\kappa ,j\in \mathbb{Z}$ with $\kappa >j$,

$$\begin{array}{cc}& \underset{B\subset {\mathbb{R}}^n}{sup}\frac{1}{{∥{\chi }_B∥}_{L^{p(·)}}}{∥{\left(b-b_B\right)}^{\upsilon }.{\chi }_B∥}_{L^{p(·)}}\approx C{\parallel b\parallel }_{*}^{\upsilon },\hfill \\ & {∥{\left(b-b_{B_j}\right)}^{\upsilon }{\chi }_{B_{\kappa }}∥}_{L^{p(·)}}\lesssim {(\kappa -j)}^{\upsilon }{\parallel b\parallel }_{*}^{\upsilon }{∥{\chi }_{B_{\kappa }}∥}_{L^{p(·)}}.\hfill \end{array}$$

The following Lemma 7 is due to Wang [38].

Lemma 7.

Let $p(·)\in {\mathcal{B}}^{*}\left({\mathbb{R}}^n\right).$ Then for any $b\in BMO\left({\mathbb{R}}^n\right)$ and $\hslash \in L^{p(·)},$ the commutator $\left[b,S_{\beta }\right]$ is bounded on $L^{p(·)}$.

Proof. 

Let $b\in BMO\left({\mathbb{R}}^n\right)$ and $\hslash \in {\dot{K}}_{p(·)}^{\alpha ,q_1(·)}\left({\mathbb{R}}^n\right)$. We can write $\hslash \left(x\right)={\sum }_{j=-\infty }^{\infty }\hslash \left(x\right){\chi }_j\left(x\right)={\sum }_{j=-\infty }^{\infty }{\hslash }_j\left(x\right).$ By the definition of ${\dot{K}}_{p(·)}^{\alpha ,q(·)}\left({\mathbb{R}}^n\right)$, we have

$${∥[b,S_{\beta }](\hslash )∥}_{{\dot{K}}_{p(·)}^{\alpha ,q_2(·)}\left({\mathbb{R}}^n\right)}=inf\left\{\vartheta >0:\sum _{l=-\infty }^{\infty }{∥{\left(\frac{2^{l\alpha }\left|[b,S_{\beta }](\hslash ){\chi }_l\right|}{\vartheta }\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}\le 1\right\}.$$

Since

$$\begin{array}{cc}\hfill {∥{\left(\frac{2^{l\alpha }\left|[b,S_{\beta }](\hslash ){\chi }_l\right|}{\vartheta }\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}& \le {∥{\left(\frac{2^{l\alpha }\left|{\sum }_{j=-\infty }^{\infty }[b,S_{\beta }]\left({\hslash }_j\right){\chi }_l\right|}{{\vartheta }_{31}+{\vartheta }_{32}+{\vartheta }_{33}}\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}\hfill \\ & \le C{∥{\left(\frac{2^{l\alpha }\left|{\sum }_{j=-\infty }^{l-3}[b,S_{\beta }]\left({\hslash }_j\right){\chi }_l\right|}{{\vartheta }_{31}}\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}\hfill \\ & +C{∥{\left(\frac{2^{l\alpha }\left|{\sum }_{j=l-2}^{l+2}[b,S_{\beta }]\left({\hslash }_j\right){\chi }_l\right|}{{\vartheta }_{32}}\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}\hfill \\ & +C{∥{\left(\frac{2^{l\alpha }\left|{\sum }_{j=l+2}^{\infty }[b,S_{\beta }]\left({\hslash }_j\right){\chi }_l\right|}{{\vartheta }_{33}}\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}},\hfill \end{array}$$

let

$$\begin{array}{cc}& {\vartheta }_{31}={∥{\left\{2^{l\alpha }\left|\sum _{j=-\infty }^{l-3}[b,S_{\beta }]\left({\hslash }_j\right){\chi }_l\right|\right\}}_{l=-\infty }∥}_{{\ell }^{q_2(·)}\left(L^{p(·)}\right)},\hfill \\ & {\vartheta }_{32}={∥{\left\{2^{l\alpha }\left|\sum _{j=l-2}^{l+2}[b,S_{\beta }]\left({\hslash }_j\right){\chi }_l\right|\right\}}_{l=-\infty }^{\infty }∥}_{{\ell }^{q_2(·)}\left(L^{p(·)}\right)},\hfill \\ & {\vartheta }_{33}={∥{\left\{2^{l\alpha }\left|\sum _{j=l+3}^{\infty }[b,S_{\beta }]\left({\hslash }_j\right){\chi }_l\right|\right\}}_{l=-\infty }^{\infty }∥}_{{\ell }^{q_2(·)}\left(L^{p(·)}\right)},\hfill \end{array}$$

and $\vartheta :=({\vartheta }_{31}+{\vartheta }_{32}+{\vartheta }_{33}):={\sum }_{\varrho =1}^3{\vartheta }_{3\varrho }$. That is,

$$\sum _{l=-\infty }^{\infty }{∥{\left(\frac{2^{l\alpha }\left|[b,S_{\beta }](\hslash ){\chi }_l\right|}{\vartheta }\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}\le C.$$

Therefore, once we prove that

$${\vartheta }_{31}\le C{∥b∥}_{*}{∥\hslash ∥}_{{\dot{K}}_{p(·)}^{\alpha ,q_1(·)}\left({\mathbb{R}}^n\right)},{\vartheta }_{32}\le C{∥b∥}_{*}{∥\hslash ∥}_{{\dot{K}}_{p(·)}^{\alpha ,q_1(·)}\left({\mathbb{R}}^n\right)},{\vartheta }_{33}\le C{∥b∥}_{*}{∥\hslash ∥}_{{\dot{K}}_{p(·)}^{\alpha ,q_1(·)}\left({\mathbb{R}}^n\right)},$$

we are done. Let ${\vartheta }_{20}={\parallel \hslash \parallel }_{{\dot{K}}_{p(·)}^{\alpha ,q_1(·)}\left({\mathbb{R}}^n\right)}$.

First, we consider ${\vartheta }_{32}$. By the $L^{p(·)}$-boundedness of $[b,S_{\beta }]$ (Lemma 7), as discussed about ${\vartheta }_{12}$ in the proof of Theorem 1, we obtain

$$\sum _{l=-\infty }^{\infty }{∥{\left(\frac{2^{l\alpha }\left|{\sum }_{j=l-2}^{l+2}[b,S_{\beta }]\left({\hslash }_j\right){\chi }_l\right|}{{\vartheta }_{20}}\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}\le C.$$

That is,

$${\vartheta }_{32}\le C{∥b∥}_{*}{\vartheta }_{20}:=C{∥b∥}_{*}{\parallel \hslash \parallel }_{{\dot{K}}_{p(·)}^{\alpha ,q_1(·)}\left({\mathbb{R}}^n\right)}.$$

Now, we estimate ${\vartheta }_{31}$. Noting that $x\in {\Re }_j,j\le l-3$. Similar to the estimation of $S_{\beta }\left({\hslash }_j\right)\left(x\right)$ in the proof of Theorem 2, we can obtain

$$S_{\beta }\left({\hslash }_j\right)\left(x\right)\lesssim 2^{-nl}{\parallel {\hslash }_j\parallel }_{L^1\left({\mathbb{R}}^n\right)}.$$

Thus,

$$[b,S_{\beta }]\left({\hslash }_j\right)\left(x\right):=|S_{\beta }\left[(b\left(x\right)-b){\hslash }_j\right]\left(x\right)|\lesssim 2^{-nl}{\parallel (b(·)-b){\hslash }_j\parallel }_{L^1\left({\mathbb{R}}^n\right)}.$$

Hence, by Lemma 3 and Hölder’s inequality (4), we get

$$\begin{array}{cc}& \sum _{l=-\infty }^{\infty }{∥{\left(\frac{2^{l\alpha }\left|{\sum }_{j=-\infty }^{l-3}[b,S_{\beta }]\left({\hslash }_j\right){\chi }_l\right|}{{\vartheta }_{20}{\parallel b\parallel }_{*}}\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}\hfill \\ & \le C\sum _{l=-\infty }^{\infty }{∥\frac{\left.2^{l\alpha }\left|{\sum }_{j=-\infty }^{l-3}\right.2^{-nl}{\parallel (b(·)-b){\hslash }_j\parallel }_{L^1\left({\mathbb{R}}^n\right)}{\chi }_l\right|}{{\vartheta }_{20}{\parallel b\parallel }_{*}}∥}_{L^{p(·)}}^{{\left(q_2^1\right)}_l}\hfill \\ & \le C\sum _{l=-\infty }^{\infty }{\left(\sum _{j=-\infty }^{l-3}{2}^{l\alpha }{∥\frac{\left|{\hslash }_j\right|}{{\vartheta }_{20}}∥}_{L^{p(·)}}{∥\left(b-b_j\right){\chi }_{B_j}∥}_{L^{p^{\prime }(·)}}\frac{2^{-nl}}{{\parallel b\parallel }_{*}}{∥{\chi }_l∥}_{L^{p(·)}}\right)}^{{\left(q_2^1\right)}_l}\hfill \\ & +C\sum _{l=-\infty }^{\infty }{\left(\sum _{j=-\infty }^{l-3}{2}^{l\alpha }{∥\frac{\left|{\hslash }_j\right|}{{\vartheta }_{20}}∥}_{L^{p(·)}}{∥\left(b-b_j\right){\chi }_{B_l}∥}_{L^{p(·)}}\frac{2^{-nl}}{{\parallel b\parallel }_{*}}{∥{\chi }_j∥}_{L^{p^{\prime }(·)}}\right)}^{{\left(q_2^1\right)}_l},\hfill \end{array}$$

where

$${\left(q_2^1\right)}_l=\left\{\begin{array}{c}{\left(q_2\right)}_{-}\mathrm{if}{∥\frac{2^{l\alpha }\left|{\sum }_{j=-\infty }^{l-3}[b,S_{\beta }]\left({\hslash }_j\right)\right|{\chi }_l}{{\vartheta }_{20}{\parallel b\parallel }_{*}}∥}_{L^{p(·)}}\le 1,\hfill \\ {\left(q_2\right)}_{+}\mathrm{otherwise}.\hfill \end{array}\right.$$

It follows from Lemmas 1–3 and 6 that

$$\begin{array}{cc}& \sum _{l=-\infty }^{\infty }{∥{\left(\frac{2^{l\alpha }\left|{\sum }_{j=-\infty }^{l-3}[b,S_{\beta }]\left({\hslash }_j\right){\chi }_l\right|}{{\vartheta }_{20}{\parallel b\parallel }_{*}}\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}\hfill \\ & \le C\sum _{l=-\infty }^{\infty }{\left(\sum _{j=-\infty }^{l-3}{2}^{-j\alpha }{∥\frac{2^{j\alpha }\left|{\hslash }_j\right|}{{\vartheta }_{20}}∥}_{L^{p(·)}}(l-j)2^{l\alpha }\frac{{∥{\chi }_{B_j}∥}_{L^{p^{\prime }(·)}}}{{∥{\chi }_{B_l}∥}_{L^{p_1^{\prime }(·)}}}\frac{\left|B_l\right|}{2^{nl}}\right)}^{{\left(q_2^1\right)}_l}\hfill \end{array}$$

$$\begin{array}{cc}& \le C\sum _{l=-\infty }^{\infty }{\left(\sum _{j=-\infty }^{l-3}{∥{\left(\frac{2^{j\alpha }\left|\hslash {\chi }_j\right|}{{\vartheta }_{20}}\right)}^{q_1(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}^{\frac{1}{{\left(q_1\right)}_{+}}}(l-j)2^{(l-j)(\alpha -n{\delta }_{11})}\right)}^{{\left(q_2^1\right)}_l}\hfill \\ & \le C\left\{\begin{array}{c}{\left\{{\displaystyle \sum _{l=-\infty }^{\infty }}{\displaystyle \sum _{j=-\infty }^{l-3}}{∥{\left(\frac{2^{j\alpha }\left|\hslash {\chi }_j\right|}{{\vartheta }_{20}}\right)}^{q_1(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}{(l-j)}^{{\left(q_1\right)}_{+}}{2}^{\frac{1}{2}(l-j)(\alpha -n{\delta }_{11}){\left(q_1\right)}_{+}}\right\}}^{q_{*}^1},{\left(q_1\right)}_{+}>1,\hfill \\ \\ {\left\{{\displaystyle \sum _{l=-\infty }^{\infty }}{\displaystyle \sum _{j=-\infty }^{l-3}}{∥{\left(\frac{2^{j\alpha }\left|\hslash {\chi }_j\right|}{{\vartheta }_{20}}\right)}^{q_1(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}{(l-j)}^{{\left(q_1\right)}_{+}}{2}^{(l-j)(\alpha -n{\delta }_{11}){\left(q_1\right)}_{+}}\right\}}^{q_{*}^1},0<{\left(q_1\right)}_{+}\le 1,\hfill \end{array}\right.\hfill \\ & \le C,\hfill \end{array}$$

where $q_{*}^1={inf}_{l\in \mathbb{N}}{\left(q_2^1\right)}_l/{\left(q_1\right)}_{+}$. This implies that

$${\vartheta }_{31}\le C{∥b∥}_{*}{\vartheta }_{20}:=C{∥b∥}_{*}{\parallel \hslash \parallel }_{{\dot{K}}_{p(·)}^{\alpha ,q_1(·)}\left({\mathbb{R}}^n\right)}.$$

Finally, we estimate ${\vartheta }_{32}$. Let $x\in {\Re }_j,j\ge l+3.$ Similar to the estimation of $S_{\beta }\left({\hslash }_j\right)$ in the proof of Theorem 2, we can find that

$$S_{\beta }\left({\hslash }_j\right)\left(x\right)\lesssim 2^{-jn}{∥{\hslash }_j∥}_{L^1\left({\mathbb{R}}^n\right)}.$$

By using the inequality stated above, we can conclude

$$[b,S_{\beta }]\left({\hslash }_j\right)\left(x\right):=|S_{\beta }\left[(b\left(x\right)-b){\hslash }_j\right]\left(x\right)|\lesssim 2^{-nj}{\parallel (b(·)-b){\hslash }_j\parallel }_{L^1\left({\mathbb{R}}^n\right)}.$$

Therefore, when $\alpha >-n{\delta }_{12}$, as argued for ${\vartheta }_{31}$, it follows that

$$\begin{array}{cc}& \sum _{l=-\infty }^{\infty }{∥{\left(\frac{2^{l\alpha }\left|{\sum }_{j=l+3}^{\infty }[b,S_{\beta }]\left({\hslash }_j\right){\chi }_l\right|}{{\vartheta }_{20}{\parallel b\parallel }_{*}}\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}\hfill \\ & \le C\sum _{l=-\infty }^{\infty }{∥\frac{\left.2^{l\alpha }\left|{\sum }_{j=l+3}^{\infty }\right.2^{-nj}{\parallel (b(·)-b){\hslash }_j\parallel }_{L^1\left({\mathbb{R}}^n\right)}{\chi }_l\right|}{{\vartheta }_{20}{\parallel b\parallel }_{*}}∥}_{L^{p(·)}}^{{\left(q_2^2\right)}_l}\hfill \\ & \le C\sum _{l=-\infty }^{\infty }{\left(\sum _{j=l+3}^{\infty }{2}^{l\alpha }{∥\frac{\left|{\hslash }_j\right|}{{\vartheta }_{20}}∥}_{L^{p(·)}}{∥\left(b-b_j\right){\chi }_{B_j}∥}_{L^{p^{\prime }(·)}}\frac{2^{-nl}}{{\parallel b\parallel }_{*}}{∥{\chi }_l∥}_{L^{p(·)}}\right)}^{{\left(q_2^2\right)}_l}\hfill \\ & +C\sum _{l=-\infty }^{\infty }{\left(\sum _{j=l+3}^{\infty }{2}^{l\alpha }{∥\frac{\left|{\hslash }_j\right|}{{\vartheta }_{20}}∥}_{L^{p(·)}}{∥\left(b-b_j\right){\chi }_{B_l}∥}_{L^{p(·)}}\frac{2^{-nl}}{{\parallel b\parallel }_{*}}{∥{\chi }_j∥}_{L^{p^{\prime }(·)}}\right)}^{{\left(q_2^1\right)}_l}\hfill \\ & \le C\sum _{l=-\infty }^{\infty }{\left(\sum _{j=l+3}^{\infty }{2}^{-j\alpha }{∥\frac{2^{j\alpha }\left|{\hslash }_j\right|}{{\vartheta }_{20}}∥}_{L^{p(·)}}(l-j)2^{l\alpha }\frac{{∥{\chi }_{B_j}∥}_{L^{p^{\prime }(·)}}}{{∥{\chi }_{B_l}∥}_{L^{p^{\prime }(·)}}}\frac{\left|B_l\right|}{2^{nl}}\right)}^{{\left(q_2^2\right)}_l}\hfill \\ & \le C\sum _{l=-\infty }^{\infty }{\left(\sum _{j=l+3}^{\infty }{∥{\left(\frac{2^{j\alpha }\left|\hslash {\chi }_j\right|}{{\vartheta }_{20}}\right)}^{q_1(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}^{\frac{1}{{\left(q_1\right)}_{+}}}(l-j)2^{(l-j)(\alpha -n{\delta }_{12})}\right)}^{{\left(q_2^2\right)}_l}\hfill \\ & \le C,\hfill \end{array}$$

where

$${\left(q_2^1\right)}_l=\left\{\begin{array}{c}{\left(q_2\right)}_{-}\mathrm{if}{∥\frac{2^{l\alpha }\left|{\sum }_{j=l+3}^{\infty }[b,S_{\beta }]\left({\hslash }_j\right)\right|{\chi }_l}{{\vartheta }_{20}{\parallel b\parallel }_{*}}∥}_{L^{p(·)}}\le 1,\hfill \\ {\left(q_2\right)}_{+}\mathrm{otherwise}.\hfill \end{array}\right.$$

$${\vartheta }_{32}\le C{\vartheta }_{20}:=C{\parallel \hslash \parallel }_{{\dot{K}}_{p(·)}^{\alpha ,q_1(·)}\left({\mathbb{R}}^n\right)}$$

This completes the proof of Theorem 3. □

3.4. Vector-Valued Inequalities for Sublinear Operators

In this section, we shall establish the boundedness of vector-valued sublinear operators on ${\dot{K}}_{p_{}(·)}^{\alpha ,q_{}(·)}\left({\mathbb{R}}^n\right)$. The boundedness of this operator on the Herz spaces with one variable $p_{}(·)$ exponents is well known [42]. For further research about the sublinear operators on different spaces with variable exponents, we refer readers to [43,44,45,46] and so on.

Theorem 4.

Suppose that $1<r<\infty ,q_1(·)\mathrm{and}{q}_2(·)\in {\mathcal{P}}_0^{*}\left({\mathbb{R}}^n\right)$ with ${\left(q_1\right)}_{+}\le {\left(q_2\right)}_{-}$ and let $p(·)\in {\mathcal{B}}^{*}\left({\mathbb{R}}^n\right),\alpha \in (-n{\delta }_{11},n{\delta }_{12})$, where ${\delta }_{11},{\delta }_{12}\in (0,1)$ are constants stated in Lemma 2. If a sublinear operator Λ verifying the size condition

$$\begin{array}{c}\hfill \left|\mathsf{\Lambda }(\hslash )\left(x\right)\right|\lesssim {\int }_{{\mathbb{R}}^n}{|x-y|}^{-n}|\hslash \left(y\right)|\mathrm{d}y,x\notin supp\hslash ,\end{array}$$

(17)

for any $\hslash \in L^1\left({\mathbb{R}}^n\right)$ with compact support and the vector-valued inequality on $L^{p(·)}$

$$\begin{array}{c}\hfill {∥{\left(\sum _{ı}{\left|\mathsf{\Lambda }\left({\hslash }_{ı}\right)\right|}^r\right)}^{\frac{1}{r}}∥}_{L^{p(·)}}\le C{∥{\left(\sum _{ı}{\left|{\hslash }_{ı}\right|}^r\right)}^{\frac{1}{r}}∥}_{L^{p(·)}},\end{array}$$

(18)

then, for any ${\left\{{\hslash }_{ı}\right\}}_{ı\in \mathbb{N}}$ satisfying ${∥{∥{\left\{{\hslash }_{ı}\right\}}_{ı}∥}_{{\ell }^r}∥}_{K_{p(·)}^{\alpha ,q_1(·)}\left({\mathbb{R}}^n\right)}<\infty ,$ the following vector-valued inequality holds:

$${∥{\left(\sum _{ı}{\left|\mathsf{\Lambda }\left({\hslash }_{ı}\right)\right|}^r\right)}^{\frac{1}{r}}∥}_{{\dot{K}}_{p(·)}^{\alpha ,q_2(·)}}\le C{∥{\left(\sum _{ı}{\left|{\hslash }_{ı}\right|}^r\right)}^{\frac{1}{r}}∥}_{{\dot{K}}_{p(·)}^{\alpha ,q_1(·)}}.$$

Proof. 

For any ${\left\{{\hslash }_{ı}\right\}}_{ı\in \mathbb{N}}$ satisfies ${∥{∥{\left\{{\hslash }_{ı}\right\}}_{ı}∥}_{{\ell }^r}∥}_{{\dot{K}}_{p(·)}^{\alpha ,q_1(·)}}<\infty ,$ we can write ${\hslash }_{ı}={\sum }_{j=-\infty }^{\infty }{\hslash }_{ı}{\chi }_j={\sum }_{j=-\infty }^{\infty }{\hslash }_{ı}^j.$ By the definition of ${\dot{K}}_{p(·)}^{\alpha ,q(·)}\left({\mathbb{R}}^n\right)$, we have

$${∥{∥{\left\{\mathsf{\Lambda }\left({\hslash }_{ı}\right)\right\}}_{ı}∥}_{{\ell }^r}{\chi }_l∥}_{{\dot{K}}_{p(·)}^{\alpha ,q_2(·)}\left({\mathbb{R}}^n\right)}=inf\left\{\vartheta >0:\sum _{l=-\infty }^{\infty }{∥{\left(\frac{2^{l\alpha }{∥{\left\{\mathsf{\Lambda }\left({\hslash }_{ı}\right)\right\}}_{ı}∥}_{{\ell }^r}{\chi }_l}{\vartheta }\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}\le 1\right\}$$

Since

$$\begin{array}{cc}\hfill {∥{\left(\frac{2^{l\alpha }{∥{\left\{\mathsf{\Lambda }\left({\hslash }_{ı}\right)\right\}}_{ı}∥}_{{\ell }^r}{\chi }_l}{\vartheta }\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}\le & {∥{\left(\frac{2^{l\alpha }{∥{\left\{{\sum }_{j=-\infty }^{\infty }\mathsf{\Lambda }\left({\hslash }_{ı}^j\right)\right\}}_{ı}∥}_{{\ell }^r}{\chi }_l}{{\vartheta }_{41}+{\vartheta }_{42}+{\vartheta }_{43}}\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}\hfill \\ \hfill \le & C{∥{\left(\frac{2^{l\alpha }{∥{\left\{{\sum }_{j=-\infty }^{l-3}\mathsf{\Lambda }\left({\hslash }_{ı}^j\right)\right\}}_{ı}∥}_{{\ell }^r}{\chi }_l}{{\vartheta }_{41}}\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}\hfill \\ & +C{∥{\left(\frac{2^{l\alpha }{∥{\left\{{\sum }_{j=l-2}^{l+2}\mathsf{\Lambda }\left({\hslash }_{ı}^j\right)\right\}}_{ı}∥}_{{\ell }^r}{\chi }_l}{{\vartheta }_{42}}\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}\hfill \end{array}$$

$$\begin{array}{cc}& +C{∥{\left(\frac{2^{l\alpha }{∥{\left\{{\sum }_{j=l+3}^{\infty }\mathsf{\Lambda }\left({\hslash }_{ı}^j\right)\right\}}_{ı}∥}_{{\ell }^r}{\chi }_l}{{\vartheta }_{43}}\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}},\hfill \end{array}$$

here

$$\begin{array}{cc}& {\vartheta }_{41}={∥{\left\{2^{l\alpha }{∥{\left\{\sum _{j=-\infty }^{l-3}\mathsf{\Lambda }\left({\hslash }_{ı}^j\right)\right\}}_{ı}∥}_{{\ell }^r}{\chi }_l\right\}}_{l=-\infty }^{\infty }∥}_{{\ell }^{q_2(·)}\left(L^{p(·)}\right)},\hfill \\ & {\vartheta }_{42}={∥{\left\{2^{l\alpha }{∥{\left\{\sum _{j=l-2}^{l+2}\mathsf{\Lambda }\left({\hslash }_{ı}^j\right)\right\}}_{ı}∥}_{{\ell }^r}{\chi }_l\right\}}_{l=-\infty }^{\infty }∥}_{{\ell }^{q_2(·)}\left(L^{p(·)}\right)},\hfill \\ & {\vartheta }_{43}={∥{\left\{2^{l\alpha }{∥{\left\{\sum _{j=l+2}^{\infty }\mathsf{\Lambda }\left({\hslash }_{ı}^j\right)\right\}}_{ı}∥}_{{\ell }^r}{\chi }_l\right\}}_{l=-\infty }^{\infty }∥}_{{\ell }^{q_2(·)}\left(L^{p(·)}\right)},\hfill \end{array}$$

and $\vartheta :=({\vartheta }_{41}+{\vartheta }_{42}+{\vartheta }_{43}):={\sum }_{\varrho =1}^3{\vartheta }_{4\varrho }$, thus,

$$\sum _{l=-\infty }^{\infty }{∥{\left(\frac{2^{l\alpha }{∥{\left\{\mathsf{\Lambda }\left({\hslash }_{ı}\right)\right\}}_{ı}∥}_{{\ell }^r}{\chi }_l}{\vartheta }\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}\le C.$$

Then,

$${∥{∥{\left\{\mathsf{\Lambda }\left({\hslash }_{ı}\right)\right\}}_{ı}∥}_{{\ell }^r}∥}_{{\dot{K}}_{p(·)}^{\alpha ,q_2(·)}\left({\mathbb{R}}^n\right)}\le C\vartheta :=C\left(\sum _{\varrho =1}^3{\vartheta }_{4\varrho }\right).$$

Hence, if we establish that

$${\vartheta }_{41}\le C{∥{∥{\left\{{\hslash }_{ı}\right\}}_{ı}∥}_{{\ell }^r}∥}_{{\dot{K}}_{p(·)}^{\alpha ,q_1(·)}\left({\mathbb{R}}^n\right)},{\vartheta }_{42}\le C{∥{∥{\left\{{\hslash }_{ı}\right\}}_{ı}∥}_{{\ell }^r}∥}_{{\dot{K}}_{p(·)}^{\alpha ,q_1(·)}\left({\mathbb{R}}^n\right)},{\vartheta }_{43}\le C{∥{∥{\left\{{\hslash }_{ı}\right\}}_{ı}∥}_{{\ell }^r}∥}_{{\dot{K}}_{p(·)}^{\alpha ,q_1(·)}\left({\mathbb{R}}^n\right)},$$

we are done. Let ${\vartheta }_0={\parallel {∥{\left\{{\hslash }_{ı}\right\}}_{ı}∥}_{{\ell }^r}\parallel }_{{\dot{K}}_{p(·)}^{\alpha ,q_1(·)}\left({\mathbb{R}}^n\right)}$.

First, we consider ${\vartheta }_{42}$. Since $\mathsf{\Lambda }$ satisfies (18), then by using the same approach as ${\vartheta }_{12}$ in the proof of Theorem 1, we can easily obtain

$$\sum _{l=-\infty }^{\infty }{∥{\left(\frac{2^{l\alpha }{∥{\left\{{\sum }_{j=l-2}^{l+2}\mathsf{\Lambda }\left({\hslash }_{ı}^j\right)\right\}}_{ı}∥}_{{\ell }^r}{\chi }_l}{\vartheta }\right)}^{q_2(·)}∥}_{L^{\frac{p(·)}{q_2(·)}}}\le C.$$

That is, ${\vartheta }_{42}\le C{\vartheta }_0:=C{∥{∥{\left\{{\hslash }_{ı}\right\}}_{ı}∥}_{{\ell }^r}∥}_{{\dot{K}}_{p(·)}^{\alpha ,q_1(·)}\left({\mathbb{R}}^n\right)}$.

Now, we estimate ${\vartheta }_{42}$. For all $l\in \mathbb{Z},j\le l-3$ and $x\in {\Re }_l$, since $\mathsf{\Lambda }$ satisfies (17), by applying Minkowski’s inequality and the inequality (4), it follows that

$$\begin{array}{cc}\hfill {∥{\left\{\mathsf{\Lambda }\left({\hslash }_{ı}^j\right)\left(x\right)\right\}}_{ı}∥}_{{\ell }^r}·{\chi }_l& \lesssim {∥{\left\{{\int }_{{\Re }_j}\left|{\hslash }_{ı}\left(y\right)\right|{|x-y|}^{-n}\mathrm{d}y\right\}}_{ı}∥}_{{\ell }^r}·{\chi }_l\hfill \\ & \lesssim 2^{-nl}{∥{\left\{{\int }_{{\Re }_j}\left|{\hslash }_{ı}\left(y\right)\right|dy\right\}}_{ı}∥}_{{\ell }^r}·{\chi }_l\hfill \\ & \lesssim 2^{-nl}{∥{∥{\left\{{\hslash }_{ı}^j\right\}}_{ı}∥}_{{\ell }^r}∥}_{L^{p(·)}}{∥{\chi }_j∥}_{L^{p^{\prime }(·)}}·{\chi }_l.\hfill \end{array}$$

From Lemmas 1–3, it follows that

$$\begin{array}{cc}& \sum _{l=-\infty }^{\infty }∥{\left(\frac{2^{l\alpha }\parallel {\sum }_{j=-\infty }^{l-3}\left\{\mathsf{\Lambda }\left({\hslash }_{ı}^j\right)\right.{\}}_{ı}{\parallel }_{{\ell }^r}{\chi }_l}{{\vartheta }_0}\right)}^{q_2(·)}∥\hfill \\ & \le C\sum _{l=-\infty }^{\infty }{\left(\sum _{j=-\infty }^{l-3}{∥{∥{\left\{\frac{{\hslash }_{ı}^j}{{\vartheta }_0}\right\}}_{ı}∥}_{{\ell }^r}∥}_{L^{p(·)}\left({\mathbb{R}}^n\right)}{2}^{l\alpha }\frac{{∥{\chi }_j∥}_{L^{p^{\prime }(·)}}}{{∥{\chi }_l∥}_{L^{p^{\prime }(·)}}}\frac{|B_l|}{2^{nl}}\right)}^{{\left(q_2^1\right)}_l}\hfill \\ & \le C\sum _{l=-\infty }^{\infty }{\left\{\sum _{j=-\infty }^{l-3}{∥{\left(\frac{2^{j\alpha }∥{\left\{{\hslash }_{ı}^{}\right\}}_{ı}{\parallel }_{{\ell }^r}{\chi }_j}{{\vartheta }_0}\right)}^{q_1(·)}∥}_{L^{\frac{p(·)}{q_1(·)}}}^{\frac{1}{{\left(q_1\right)}_{+}}}{2}^{(l-j)(\alpha -{\delta }_{12})}\right\}}^{{\left(q_2^1\right)}_l},\hfill \end{array}$$

where

$${\left(q_2^1\right)}_l=\left\{\begin{array}{c}{\left(q_2\right)}_{-}\mathrm{if}{∥\frac{2^{l\alpha }\left|{\sum }_{j=-\infty }^{l-3}\mathsf{\Lambda }\left({\hslash }_j\right)\right|{\chi }_l}{{\vartheta }_{20}}∥}_{L^{p(·)}}\le 1,\hfill \\ {\left(q_2\right)}_{+}\mathrm{otherwise}.\hfill \end{array}\right.$$

Moreover, using the same approach as ${\vartheta }_{11}$ in the proof of Theorem 1, we find

$$\begin{array}{cc}& \sum _{l=-\infty }^{\infty }∥{\left(\frac{2^{l\alpha }\parallel {\sum }_{j=-\infty }^{l-3}\left\{\mathsf{\Lambda }\left({\hslash }_{ı}^j\right)\right.{\}}_{ı}{\parallel }_{{\ell }^r}{\chi }_l}{{\vartheta }_0}\right)}^{q_2(·)}∥\le C.\hfill \end{array}$$

Finally, we estimate ${\vartheta }_{43}$. For all $l\in \mathbb{Z},j\le l-3$ and $x\in {\Re }_l$, since $\mathsf{\Lambda }$ satisfies (17), by applying Minkowski’s inequality and the inequality (4), it follows that

$$\begin{array}{cc}\hfill {∥{\left\{\mathsf{\Lambda }\left({\hslash }_{ı}^j\right)\left(x\right)\right\}}_{ı}∥}_{{\ell }^r}·{\chi }_l& \lesssim {∥{\left\{{\int }_{{\Re }_j}\left|{\hslash }_{ı}\left(y\right)\right|{|x-y|}^{-n}\mathrm{d}y\right\}}_{ı}∥}_{{\ell }^r}·{\chi }_l\hfill \\ & \lesssim 2^{-nj}{∥{\left\{{\int }_{{\Re }_j}\left|{\hslash }_{ı}\left(y\right)\right|dy\right\}}_{ı}∥}_{{\ell }^r}·{\chi }_l\hfill \\ & \lesssim 2^{-nj}{∥{∥{\left\{{\hslash }_{ı}^j\right\}}_{ı}∥}_{{\ell }^r}∥}_{L^{p(·)}}{∥{\chi }_j∥}_{L^{p^{\prime }(·)}}·{\chi }_l.\hfill \end{array}$$

Thus, when $\alpha >-n{\delta }_{11}$, similar to the approach taken to estimate ${\vartheta }_{41}$ before, we can obtain

$$\begin{array}{cc}& \sum _{l=-\infty }^{\infty }∥{\left(\frac{2^{l\alpha }\parallel {\sum }_{j=l+3}^{\infty }\left\{\mathsf{\Lambda }\left({\hslash }_{ı}^j\right)\right.{\}}_{ı}{\parallel }_{{\ell }^r}{\chi }_l}{{\vartheta }_0}\right)}^{q_2(·)}∥\hfill \\ \hfill \le & C\sum _{l=-\infty }^{\infty }{∥\frac{2^{l\alpha }\parallel {\sum }_{j=l+3}^{\infty }\left\{\mathsf{\Lambda }\left({\hslash }_{ı}^j\right)\right.{\}}_{ı}{\parallel }_{{\ell }^r}{\chi }_l}{{\vartheta }_0}∥}_{L^{\frac{p(·)}{q_2(·)}}}^{{\left(q_2^2\right)}_l}\hfill \\ \hfill \le & C\sum _{l=-\infty }^{\infty }{\left\{\sum _{j=l+3}^{\infty }{∥{∥{\left\{\frac{{\hslash }_{ı}^j}{{\vartheta }_0}\right\}}_{ı}∥}_{{\ell }^r}∥}_{L^{p(·)}}{2}^{l\alpha }\frac{{∥{\chi }_l∥}_{L^{p(·)}}}{{∥{\chi }_j∥}_{L^{p(·)}}}\frac{|B_j|}{2^{nj}}\right\}}^{{\left(q_2^2\right)}_l}\hfill \\ \hfill \le & C\sum _{l=-\infty }^{\infty }{\left\{\sum _{j=l+3}^{\infty }{∥{\left(\frac{2^{j\alpha }∥{\left\{{\hslash }_{ı}^{}\right\}}_{ı}{\parallel }_{{\ell }^r}{\chi }_j}{{\vartheta }_0}\right)}^{q_1(·)}∥}_{L^{\frac{p(·)}{q_1(·)}}}^{\frac{1}{{\left(q_1\right)}_{+}}}{2}^{(l-j)(\alpha +{\delta }_{11})}\right\}}^{{\left(q_2^2\right)}_l}\hfill \\ \hfill \le & C.\hfill \end{array}$$

where

$${\left(q_2^2\right)}_l=\left\{\begin{array}{c}{\left(q_2\right)}_{-}\mathrm{if}{∥\frac{2^{l\alpha }∥{\sum }_{j=l+3}^{\infty }{\left\{\mathsf{\Lambda }\left({\hslash }_{ı}^j\right)\right\}}_{ı}{\parallel }_{{\ell }^r}{\chi }_l}{{\vartheta }_0}∥}_{L^{p(·)}}\le 1,\hfill \\ {\left(q_2\right)}_{+}\mathrm{otherwise}.\hfill \end{array}\right.$$

This completes the proof of Theorem 4. □

Remark 2.

(1) It is worth pointing out that even in the particular case that $q(·)$ is constant, our main results, namely Theorems 1, and 2, are also new.

(2) All of these results can also be proved for non-homogeneous Herz spaces with two variable exponents. Due to the similarity of the arguments, details are omitted.

4. Conclusions

We obtain the boundedness of the vector-valued intrinsic square function and the boundedness of the scalar-valued intrinsic square function on Herz spaces with two variable exponents. The boundedness of the corresponding commutators generated by a BMO function and scalar-valued intrinsic square function is also obtained. As a supplement, the vector-valued inequality for sublinear operators is obtained on ${\dot{K}}_{p(·)}^{\alpha ,q_{}(·)}\left({\mathbb{R}}^n\right)$.