Boundedness of fractional Marcinkiewicz integral with variable kernel on variable Morrey-Herz spaces

1. Introduction

In \(1938\) Marcinkiewicz [1] introduced the Marcinkiewicz integral. Later, Stain [2] studied the bounded of Marcinkiewicz integral on \(L^{p}(\mathbb{R}^{n})\) for any \( p\in (0,1]\) and bounded from \(L^{1}(\mathbb{R}^{n})\) to \(L^{1,\infty}(\mathbb{R}^{n})\). Torchinsky and Wang in [3] discussed the boundedness for the commutator generated by the Marcinkiewicz integral \(\mu_{\Omega}\) and \(BMO(\mathbb{R}^{n})\) function on Lebesgue spaces \(L^{p}(\mathbb{R}^{n})\). Pan and Wang [4] established the boundedness of fractional Marcinkiewicz integrals with variable kernels on Hardy type spaces.

Moving in another direction, Kov\(\acute{a}\check{c}\)ik and R\(\acute{a}\)kosn\(\acute{i}\)k [5] introduced the class of variable exponent Lebesgue spaces \(L^{p(\cdot)}(\mathbb{R}^{n})\), after that many authors has been interested in studying the function spaces with variable exponents [6, 7, 8, 9, 10, 11]. Izuki [12] and [13] introduced the class of variable exponent Herz-Morrey space \( {M\dot{K}_{q,p(\cdot)}^{\alpha,\lambda}(\mathbb{R}^{n})}\) and the boundedness of the sublinear operator satisfying size condition and fractional integral on those spaces were proved. Hongbin Wang [14] studied the commutator of Marcinkiewicz integrals on Herz spaces with variable exponent. In recent years, Yan Lu and Yue Ping [15] introduced the class of \( {M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})}\) and also established the boundedness of multilinear Calder\'{o}n-Zygmund singular operators on those spaces.

In this paper, we consider the boundedness of the fractional Marcinkiewicz integral with variable kernel \([\mu_{\Omega,\gamma}]\) and its commutators \([b^{m}, \mu_{\Omega,\gamma}]\) on \(M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})\).

Now assume that \(S^{n-1} (n\geq2)\) the unit sphere in \(\mathbb{R}^{n}\) with the normalized Lebesgue measure \( d\sigma(x')\). Let \(\Omega\in{L}^{\infty}(\mathbb{R}^{n})\times{L^{r}(S^{n-1})}(r\geq1)\) be homogeneous of degree zero and satisfy the vanishing condition on \( S^{n-1}\), that is

The fractional Marcinkiewicz integral operator with variable kernel \(\mu_{\Omega,\gamma}( 0< \gamma< n)\) is defined by where Obviously, if \(\gamma=0\), \(\mu_{\Omega,\gamma}\) is just Marcinkiewicz integral with variable kernels is defined by If \(\Omega\equiv 1\), then \(\mu_{\Omega,\gamma}=\mu_{\gamma}\) is a fractional Marcinkiewicz integral operator is defined by The commutators of the fractional Marcinkiewicz integral operator with variable kernels is defined by Now, we need the further assumption for \(\Omega(x ,z)\), the integral modulus \( \omega_{r}(\delta)\) of \( L^{r}\) continuity of \(\Omega\) is defined by $${\omega_{r}(\delta)}= \sup_{\|\mu\|< \delta } \left(\int\limits_{s^{n - 1}}|\Omega(x , \mu{z'}) - \Omega (x , z')|^{r}d\sigma(z')\right)^{\frac{1}{r}}, 1\leq r < \infty $$ and $$ \omega_{\infty}(\delta)= \sup\limits_{\|\mu\|< \delta}|\Omega(x , \mu{z'})-\Omega (x , z')|,$$ where \( 0< \delta\leq 1\), \(\mu\) is the rotation in \( s^{n-1}\), \(\|\mu\|= \sup\limits_{z'\in S^{n - 1}}|\mu{z'} - {z'}|\). Let \(E\) be a measurable set in \(\mathbb{R}^{n}\) with \(|E|> 0 \). Next, we introduce the Lebesgue spaces with variable exponents.

Definition 1.1. [5] Let \(p(\cdot): E \rightarrow {[1,\infty)}\) be a measurable function. The Lebesgue space with variable exponent \(L^{p(\cdot)}(E)\) is defined by $$ L^{p(\cdot)}(E)= \{{ f~ \mbox{is measurable}: \int\limits_{E}\left(\frac{|f(x)|}{\eta}\right)^{p(x)} dx < \infty}~ \mbox{for some constant } \eta > 0\}. $$ The space \(L _{loc}^{p(\cdot)} {(E)}\) is defined by $$ L_{loc}^{p(\cdot)} {(E)}= \{ \mbox {\(f\) is measurable}: f\in {L^{p(\cdot)} {(K)}} \mbox{ for all compact}~ K\subset{E}\}. $$ The Lebesgue spaces \(L^{p(\cdot)} {(E)}\) is a Banach spaces with the norm defined by $$ \|f\|_{L^{p(\cdot)}(E)}= \inf\{\eta> 0 : \int\limits_{E}\left(\frac{|f(x)|}{\eta}\right)^{p(x)}dx \leq 1\}. $$

We denote \(p_{-}=\) essinf \(\{p(x): x \in E\} , \) \( p_{+}=\) esssup\( \{p(x): x \in E\} \). Then \(\mathcal{P}(E)\) consists of all \(p(\cdot)\) satisfying \(p_{-} > 1\) and \(p_{+} < \infty\), and $$\mathfrak{B}(E)= \left\{ p(\cdot)\in\mathcal{P}(E) :\mbox{ M is bounded on }L^{p(\cdot)}\right\},$$ where the Hardy-Littlewood maximal operator \(M\) is defined by $$Mf(x)= \sup\limits _{r>0} r^{-n}\int\limits_{B(x,r)\cap E}|f(y)|dy.$$ We see that a function \(\alpha(\cdot): \mathbb{R}^{n} \longrightarrow \mathbb{R}\) is called log-Hölder continuous at the origin(or has a log decay at the origin ), if there exists a constant \(C_{log}> 0\) such that $$ | \alpha(x)-\alpha(0)|\leq \frac{C_{log}}{log(e + \frac{1}{|x|})},\;\;\;\;\;\;\; \forall{x}\in \mathbb{R}^{n}.$$ If for some \(\alpha_{\infty}\in\mathbb{R} \) and \(C_{log}> 0\), we have $$ | \alpha(x)-\alpha_{\infty}|\leq \frac{C_{log}}{log(e + {|x|})},\;\;\;\;\;\;\;\forall{x}\in \mathbb{R}^{n}.$$ Then \(\alpha(\cdot)\) is called log-Hölder continuous at infinity (or has a log decay at infinity).

By \( \mathcal{P}_{0}^{log}(\mathbb{R}^{n}) \) and \( \mathcal{P}_{\infty}^{log}(\mathbb{R}^{n}) \) we denote the class of all exponents \(p\in \mathcal{P}(\mathbb{R}^{n})\) which have a log decay at the origin and at infinity, respectively. It is worth noting that if \( p(\cdot)\in \mathcal{P}_{0}^{log}(\mathbb{R}^{n}) \cap \mathcal{P}_{\infty}^{log}(\mathbb{R}^{n}) \), then we have \(p(\cdot)\in \mathfrak{B}(\mathbb{R}^{n}).\)

Let \(B_{k}=\{ x\in\mathbb{R}^{n} : |x|\leq 2^{k}\}, C_{k}= B_{k}\backslash B_{k-1}, \chi_{k}= \chi_{B_{k}}, k \in{\mathbb{Z}}.\) Almeida and Direhem in [1] introduced the following definition.

Definition 1.2. Let \( 0< q \leq \infty , p(\cdot)\in \mathcal{P} (\mathbb{R}^{n})\) and \(\alpha(\cdot):\mathbb{R}^{n}\longrightarrow \mathbb{R}\) with \(\alpha \in L^{\infty}(\mathbb{R}^{n})\).

1. The homogeneous Herz space \(\dot{K}_{p(\cdot)}^{\alpha(\cdot),q}(\mathbb{R}^{n})\) is defined as the set of all \( f \in {L^{p(\cdot)}_{loc}}(\mathbb{R}^{n} \setminus\{0\})\) such that $$ \|f\|_{\dot{K}_{p(\cdot)}^{\alpha(\cdot),q}(\mathbb{R}^{n})}:= \left( \sum_{k\in \mathbb{Z}}\|2^{k\alpha(\cdot)}f\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}^{q} \right)^{1/q}< \infty.$$
2. The nonhomogeneous Herz space \({K}_{p(\cdot)}^{\alpha(\cdot),q}(\mathbb{R}^{n})\) consists of all \( f \in {L^{p(\cdot)}_{loc}}(\mathbb{R}^{n})\) such that $$ \|f\|_{{K}_{p(\cdot)}^{\alpha(\cdot),q}(\mathbb{R}^{n})}:= \|f\chi_{B_{0}}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}+\left( \sum_{k\geq 1}\|2^{k\alpha(\cdot)}f\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}^{q} \right)^{1/q}< \infty,$$ with the usual modification when \(q=\infty.\)

Definition 1.3. Let \( 0< q \leq \infty , p(\cdot)\in \mathcal{P} (\mathbb{R}^{n})\),\(0\leq\lambda< \infty\) and \(\alpha(\cdot):\mathbb{R}^{n}\longrightarrow \mathbb{R}\) with \(\alpha \in L^{\infty}(\mathbb{R}^{n})\). The homogeneous Morrey-Herz space \(M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})\) is defined as the set of all \( f \in {L^{p(\cdot)}_{loc}}(\mathbb{R}^{n} \setminus\{0\})\) such that $$ \|f\|_{M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})}:= \sup_{L\in\mathbb{Z}}2^{-L\lambda}\left( \sum_{k=-\infty}^{L}\|2^{k\alpha(\cdot)}f\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}^{q} \right)^{1/q}< \infty, $$ with the usual modification when \(q=\infty.\)

Remark 1.1. If \(\alpha(\cdot)\) is constant, then \({M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})}={M\dot{K}_{q,p(\cdot)}^{\alpha,\lambda}(\mathbb{R}^{n})}\). If \(\lambda=0\), then \({M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})}={\dot{K}_{q,p(\cdot)}^{\alpha(\cdot)}(\mathbb{R}^{n})}\). If both \(\alpha(\cdot)\) and \(p(\cdot)\) are constant and \(\lambda=0\), then \({M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})}={\dot{K}_{q,p}^{\alpha}(\mathbb{R}^{n})}\) is the classical Herz space introduced in [17].

Definition [18] For \(0< \beta \leq 1\), the Lipschitz space \(Lip_{\beta}(\mathbb{R}^{n})\) is defined by $$ Lip_{\beta}(\mathbb{R}^{n})= \left\{ f : \|f\|_{Lip_{\beta}}= \sup\limits_{x,y\in\mathbb{R}^{n}, x\neq y }\frac{|f(x)-f(y)|}{|x-y|^{\beta}} < \infty\right\}. $$

2. Properties of variable exponents

Before stating our main results, we introduce some key Lemmas which will be used later. The next Lemma is the generalization of Herz with variable exponents spaces in [16], and it was used by Lu and Zhu in [15].

Lemma 2.1. Let \(0< q \leq \infty\), \(p(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\),\(0\leq\lambda< \infty\) and \(\alpha \in L^{\infty}(\mathbb{R}^{n})\). If \(\alpha(\cdot)\) is log-Hölder continuous both at the origin and at infinity, then $$ \begin{array}{ll} \| f\|_{M\dot{K}_{q,p(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})}\approx max \left\{{\sup\limits_{k_{0}< 0,k_{0}\in \mathbb{Z}} 2^{-k_{0}\lambda } \left(\sum\limits_{k=-\infty}^{k_{0}}2^{k\alpha(0)q} \left\|f\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{1/q}}, \right.\\ \left. \sup\limits_{k_{0}\geq 0,k_{0}\in \mathbb{Z}} \left[2^{-k_{0}\lambda } \left(\sum\limits_{k=-\infty}^{-1} 2^{k\alpha(0)q}\left\|f\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{1/q}\right] \right.\\ \left. + \left[2^{-k_{0}\lambda }\left( \sum\limits_{k=0}^{k_{0}}2^{k\alpha_{\infty}q} \left\|f\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{1/q}\right]\right\}. \end{array} $$

Lemma 2.2. [10]

1. If \(p\in \mathcal{P}(\mathbb{R}^{n})\), then for all \( f \in L^{q(\cdot)}(\mathbb{R}^{n})\) and \(g \in L^{p'(\cdot)}(\mathbb{R}^{n}) \), we have $$\int\limits_{\mathbb{R}^{n}}|f(x) g(x)| dx \leq C\|f\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\|g\|_{L^{p'(\cdot)}(\mathbb{R}^{n})}.$$
2. If \( p(\cdot), q (\cdot), r(\cdot) \in \mathbb{R}^{n} \), define \( p(\cdot)\) by : \( \frac{1}{p(\cdot)}= \frac{1}{q(\cdot)} + \frac{1}{r(\cdot)}\). Then there exists a constant C such that for all \( f \in L^{q(\cdot)}(\mathbb{R}^{n}) , ~ g \in L^{r(\cdot)}(\mathbb{R}^{n}) \), we have $$\|fg\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\leq C\|f\|_{L^{q(\cdot)}(\mathbb{R}^{n})}\|g\|_{L^{r(\cdot)}(\mathbb{R}^{n})}.$$

Lemma 2.3. [19] Suppose that \(p_{1}(\cdot)\in \mathfrak{B}(\mathbb{R}^{n})\), \(\Omega\in{L}^{\infty}(\mathbb{R}^{n})\times{L^{r}(S^{n-1})}\). Let \(0< \gamma\leq\frac{n}{( p_{1})_{+}} \), and define \(p_{2}(\cdot)\) by: \(\frac{1}{p_{1}(x)} - \frac{1}{p_{2}(x)} = \frac{\gamma}{n}\), then for all \(f \in L^{p_{1}(\cdot)}(\mathbb{R}^{n})\), we have $$\|T_{\Omega , \gamma} f\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\leq C \| f\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}.$$

Lemma 2.4. [20] Suppose that \(p_{1}(\cdot)\in \mathfrak{B}(\mathbb{R}^{n})\), \( b\in Lip_{\beta}(\mathbb{R}^{n})\), \(0 < \beta\leq1\), \(\Omega\in{L}^{\infty}(\mathbb{R}^{n})\times{L^{r}(S^{n-1})}\). Let \(0< \gamma+ m\beta\leq\frac{n}{( p_{1})_{+}} \), and define \(p_{2}(\cdot)\) by : \(\frac{1}{p_{1}(x)} - \frac{1}{p_{2}(x)} = \frac{\gamma + m\beta}{n}\). Then for all \(f \in L^{p_{1}(\cdot)}(\mathbb{R}^{n})\), we have $$ \|[ b^{m} ,T_{\Omega, \gamma}] f\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\leq C \|b\|^{m}_{Lip_{\beta}} \| f\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}. $$

Lemma 2.5.

1. Suppose that \( 0< \gamma < n, p_{1}(\cdot)\in \mathfrak{B}(\mathbb{R}^{n})\), \(\Omega\in{L}^{\infty}(\mathbb{R}^{n})\times{L^{r}(S^{n-1})}\). Let \(0< \gamma\leq\frac{n}{( p_{1})_{+}} \) and define \(p_{2}(\cdot)\) by \(\frac{1}{p_{1}(x)} - \frac{1}{p_{2}(x)} = \frac{\gamma}{n}\), then for all \(f \in L^{p_{1}(\cdot)}(\mathbb{R}^{n})\), we have $$\|\mu_{\Omega ,\gamma} f\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\leq C \| f\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}.$$
2. Suppose that \(b\in~Lip(\mathbb{R}^{n})\), \( 0< \gamma < n, p_{1}(\cdot)\in \mathfrak{B}(\mathbb{R}^{n})\), \(\Omega\in{L}^{\infty}(\mathbb{R}^{n})\times{L^{r}(S^{n-1})}\). Let \(0< \gamma+m\beta\leq\frac{n}{( p_{1})_{+}} \) and define \(p_{2}(\cdot)\) by \(\frac{1}{p_{1}(x)} - \frac{1}{p_{2}(x)} = \frac{\gamma+m\beta}{n}\), then for all \(f \in L^{p_{1}(\cdot)}(\mathbb{R}^{n})\), we have $$\|[ b^{m} ,\mu_{\Omega, \gamma}] f\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\leq C \|b\|^{m}_{Lip_{\beta}} \| f\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}. $$

Proof.

1. Applying the Minkowski inequality, we get \begin{align*} \mu_{\Omega,\gamma}(f) (x)&\leq C \int\limits_{\mathbb{R}^{n}}\frac{|\Omega(x, x-y )|}{|x -y|^{n-\gamma-1}}|f(y)| \left(\int\limits_{|x-y|}^{\infty}\frac{dt}{t^{3}}\right)^{\frac{1}{2}}dy\\ &\leq C \int\limits_{\mathbb{R}^{n}}\frac{|\Omega(x, x-y )|}{|x -y|^{n-\gamma-1}}|f(y)| \frac{1}{|x-y|}dy\\ &\leq C \int\limits_{\mathbb{R}^{n}}\frac{|\Omega(x, x-y )|}{|x -y|^{n-\gamma}}|f(y)|dy. \end{align*} We know that \(T_{|\Omega|,\gamma}|f|\) is bounded on \(L^{p(\cdot)}(\mathbb{R}^{n})\) (Lemma 2.3), then we obtain \begin{align*} \|\mu_{\Omega,\gamma}(f) (x)\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} &\leq C\|T_{|\Omega|,\gamma}|f|\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\\ &\leq C \|f\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}. \end{align*}
2. By the same way which was used in above and applying Lemma 2.4 we can easily obtain the boundedness of the commutator of fractional Marcinkiewicz integral with variable kernel on Lebesgue spaces with variable exponent \(L^{p(\cdot)}(\mathbb{R}^{n})\).

Lemma 2.6. [21] If \(0< \gamma< n\) and \(p(\cdot), q(\cdot)\in \widetilde{B}\) such that \(p^{+}< \frac{n}{\gamma}\) and define $$ \frac{1}{p(x)}-\frac{1}{q(x)}=\frac{\gamma}{n}, ~~~~~~~~~ x\in \mathbb{R}^{n}, $$ we have $$ \|\chi_{B}\|_{L^{q(\cdot)}(\mathbb{R}^{n})}\sim |B|^{-\frac{\gamma}{n}}\|\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}. $$

Lemma 2.7. [8] If \(p(\cdot) \in \mathfrak{B}(\mathbb{R}^{n})\), then there exist constants \(\delta_{1},\delta_{2},C > 0 \) such that for all balls \(B\) in \(\mathbb{R}^{n}\) and all measurable subset \(S\subset B\) $$ \frac{\|\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}} {\|\chi_{S}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}} \leq C \frac{|B|}{|S|} ,~ \frac{\|\chi_{S}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})}} {\|\chi_{B}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})}} \leq C \left(% \begin{array}{ccc} \frac{|S|}{|B|}\end{array}\right)^{\delta_{1}}, ~ \frac{\|\chi_{S}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}} {\|\chi_{B}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}} \leq C \left(% \begin{array}{ccc} \frac{|S|}{|B|}\end{array}\right)^{\delta_{2}}. $$

Lemma 2.8. [6] If \(p(\cdot) \in \mathfrak{B}(\mathbb{R}^{n})\), there exists constant \(C > 0\) such that for any balls B in \(\mathbb{R}^{n}\), we have $$ \frac{1}{|B|}\|\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})} \|\chi_{B}\|_{L^{p'(\cdot)}(\mathbb{R}^{n})}\leq C. $$

Lemma 2.9.[7] Let \(b\in Lip_{\beta}(\mathbb{R}^{n})\), \(m\) be a positive integer, and there exist constants \(C> 0\), such that for any \( l,j \in\mathbb{Z}\) with \(l>j\) \( (1)~ C^{-1}\|b\|^{m}_{Lip_{\beta}}\leq |B|^{-m\beta/n}\|\chi_{B}\|^{-1}_{L^{p(\cdot)}(\mathbb{R}^{n})}\|( b- b_{B})^{m}\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})} \leq C \|b\|^{m}_{Lip_{\beta}}\) \( (2)~\|( b- b_{B_{j}})^{m}\chi_{B_{k}}\|_{L^{p(\cdot)}(\mathbb{R}^{n})} \leq C |B_{k}|^{m\beta/n} \|b\|^{m}_{Lip_{\beta}}\| \chi_{B_{k}}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}.\)

3. Main Theorems and their proofs

Theorem 3.1. Suppose that \( 0< \gamma< n,0< q\leq\infty\) and \(\Omega\in{L}^{\infty}(\mathbb{R}^{n})\times{L^{r}(S^{n-1})}\) \((1< r\leq \infty)\). Let \(p_{1}(\cdot), p_{2}(\cdot) \in \mathfrak{B}(\mathbb{R}^{n})\), \(1\leq r' < (p_{1})_{-}\leq (p_{1})_{+} < \frac{n}{\gamma}\), \(\frac{1}{p_{1}(x)} - \frac{1}{p_{2}(x)} = \frac{\gamma}{n}\) and let \(\alpha(x)\in L^{\infty}(\mathbb{R}^{n})\) be log-Hölder continuous both at the origin and at infinity, such that

where \(\delta_{1}, \delta_{2}\) are the constants in Lemma 2.7. Then \(\mu_{\Omega,\gamma}\) is bounded from \newline \({M\dot{K}_{q_{1},p_{1}(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})}\) to \({M\dot{K}_{q_{2},p_{2}(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})}\).

Proof. If \(f \in {M\dot{K}_{q_{1},p_{1}(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})}\), we applying $$\left[ \sum\limits_{j=1}^{\infty}a_{j}\right]^{q_{1}/q_{2}}\leq \sum\limits_{j=1}^{\infty}a_{j}^{q_{1}/q_{2}}, \,\,\,\,\,\,\, a_{1}, a_{2} ....\geq 0.$$ \begin{align*} \left\|\mu_{\Omega, \gamma} f_{j}\right\|^{q_{1}}_{M\dot{K}_{q_{2},p_{2}(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})}&= \sup\limits_{k_{0}\in \mathbb{Z}} 2^{-k_{0}\lambda {q_{1}}} \left\{\sum_{k=-\infty}^{k_{0}} \left\|2^{k\alpha(\cdot)}\mu_{\Omega,\gamma} (f)\chi_{k}\right\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}^{q_{2}}\right\}^{q_{1}/q_{2}}\\ &\leq \sup\limits_{k_{0}\in \mathbb{Z}} 2^{-k_{0}\lambda {q_{1}}} \sum\limits_{k=-\infty}^{k_{0}} \left\|2^{k\alpha(\cdot)}\mu_{\Omega,\gamma} (f)\chi_{k}\right\|^{q_{1}}_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}. \end{align*} If we denote $$ f(x) = \sum_{j=-\infty}^{\infty}f(x)\chi_{j}= {\sum_{j=- \infty}^{ \infty}} f_{j}(x). $$ Then, we have \begin{align*} & \left\|\mu_{\Omega,\gamma} f_{j}\right\|^{q_{1}}_{M\dot{K}_{q_{2},p_{2}(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})}\\ &\leq \sup\limits_{k_{0}\in \mathbb{Z}} 2^{-k_{0}\lambda {q_{1}}} \sum\limits_{k=-\infty}^{k_{0}} \left\|2^{k\alpha(\cdot)}\left(\sum\limits_{j=-\infty}^{\infty}|\mu_{\Omega,\gamma} f_{j}|\right)\chi_{k}\right\|^{q_{1}}_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\\ &\leq \sup\limits_{k_{0}\in \mathbb{Z}} 2^{-k_{0}\lambda {q_{1}}} \sum\limits_{k=-\infty}^{k_{0}} \left\|2^{k\alpha(\cdot)}\left(\sum\limits_{j=-\infty}^{k-2}|\mu_{\Omega,\gamma} f_{j}|\right)\chi_{k}\right\|^{q_{1}}_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\\ &\quad+ \sup\limits_{k_{0}\in \mathbb{Z}} 2^{-k_{0}\lambda {q_{1}}} \sum\limits_{k=-\infty}^{k_{0}} \left\|2^{k\alpha(\cdot)}\left(\sum\limits_{j=k-1}^{k+1}|\mu_{\Omega,\gamma} f_{j}|\right)\chi_{k}\right\|^{q_{1}}_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\\ &\quad+ \sup\limits_{k_{0}\in \mathbb{Z}} 2^{-k_{0}\lambda {q_{1}}} \sum\limits_{k=-\infty}^{k_{0}} \left\|2^{k\alpha(\cdot)}\left(\sum\limits_{j=k+2}^{\infty}|\mu_{\Omega,\gamma} f_{j}|\right)\chi_{k}\right\|^{q_{1}}_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})},\\ &L_{11}+ L_{12}+L_{13}. \end{align*} Therefore, we have

Below, we first consider \(L_{12}\). By Lemma 2.1, we get \begin{align*} L_{12}&= max \left\{{\sup\limits_{k_{0}< 0,k_{0}\in \mathbb{Z}} 2^{-k_{0}\lambda {q_{1}}} \sum\limits_{k=-\infty}^{k_{0}} \left\|2^{k\alpha(0)}\left(\sum\limits_{j=k-1}^{k+1}|\mu_{\Omega,\gamma} f_{j}|\right)\chi_{k}\right\|^{q_{1}}_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}},\right.\\ &\left.\sup\limits_{k_{0}\geq 0,k_{0}\in \mathbb{Z}} \left[2^{-k_{0}\lambda {q_{1}}} \sum\limits_{k=-\infty}^{-1} \left\|2^{k\alpha(0)}\left(\sum\limits_{j=k-1}^{k+1}|\mu_{\Omega,\gamma} f_{j}|\right)\chi_{k}\right\|^{q_{1}}_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\right.\right.\\ &\left.\left.+ 2^{-k_{0}\lambda {q_{1}}} \sum\limits_{k=0}^{k_{0}} \left\|2^{k\alpha_\infty}\left(\sum\limits_{j=k-1}^{k+1}|\mu_{\Omega,\gamma} f_{j}|\right)\chi_{k}\right\|^{q_{1}}_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\right]\right\}. \end{align*} Noting that \(\mu_{\Omega,\gamma}\) is bounded on \(L^{p(\cdot)}{(\mathbb{R}^{n})}\) (Lemma 2.5), we have \begin{align*} \left\|2^{k\alpha(0)}\left(\sum\limits_{j=k-1}^{k+1}|\mu_{\Omega,\gamma} f_{j}|\right)\chi_{k}\right\|^{q_{1}}_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} &\leq\left\|2^{k\alpha(0)}\left(\sum\limits_{j=k-1}^{k+1}|\mu_{\Omega,\gamma} f_{j}|\right)\right\|^{q_{1}}_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\\ &\leq\sum\limits_{j=k-1}^{k+1}\|2^{k\alpha(0)}|\mu_{\Omega,\gamma} f_{j}|\|^{q_{1}}_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\\ &\leq\sum\limits_{j=k-1}^{k+1}\|2^{k\alpha(0)}| f_{j}|\|^{q_{1}}_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}. \end{align*} Thus, we have Now we consider \(L_{11}\), we have \begin{align*} |\mu_{\Omega,\gamma}f_{j}(x)| &\leq \left(\int\limits_{0}^{|x|}\left|\int_{|x-y|\leq t}\frac{\Omega(x, x-y )}{|x -y|^{n-\gamma-1}}f_{j}(y)dy\right|^{2}\frac{dt}{t^{3}}\right)^{\frac{1}{2}}\\ &\quad+ \left(\int\limits_{|x|}^{\infty}\left|\int_{|x-y|\leq t}\frac{\Omega(x, x-y )}{|x -y|^{n-\gamma-1}}f_{j}(y)dy\right|^{2}\frac{dt}{t^{3}}\right)^{\frac{1}{2}}\\ &= I_{1}+ I_{2}. \end{align*} Noting that \( x \in C_{k}, ~ j \leq k-2 \), then \( | x- y|\sim |x|\sim |2^{k}|\). Hence By (10), and the Minkowski inequality, we show that \begin{align*} I_{1}&\leq C \int\limits_ {\mathbb{R}^{n}}\frac{|\Omega(x, x-y )|}{|x -y|^{n-\gamma-1}}|f_{j}(y)| \left(\int\limits_{|x-y|}^{|x|}\frac{dt}{t^{3}}\right)^{\frac{1}{2}}dy\\ &\leq C \int\limits_{ \mathbb{R}^{n}}\frac{|\Omega(x, x-y )|}{|x -y|^{n-\gamma-1}}|f_{j}(y)| \left|\frac{1}{|x-y|^{2}}-\frac{1}{|x|^{2}}\right|^{\frac{1}{2}}dy\\ &\leq C \int\limits_{ \mathbb{R}^{n}}\frac{|\Omega(x, x-y )|}{|x -y|^{n-\gamma-1}}|f_{j}(y)| \frac{|y|^{\frac{1}{2}}}{|x-y|^{\frac{3}{2}}}dy\\ &\leq C 2^{(j-k)\frac{1}{2}}\int\limits_{ C_{j}}\frac{|\Omega(x, x-y )|}{|x -y|^{n-\gamma}} |f_{j}(y)|dy\\ &\leq C \int\limits_{ C_{j}}\frac{|\Omega(x, x-y )|}{|x -y|^{n-\gamma}} |f_{j}(y)|dy. \end{align*} Similarly, we estimate \(I_{2}\). By the Minkowski inequality, we get \begin{align*} I_{2}&\leq C \int\limits_ {\mathbb{R}^{n}}\frac{|\Omega(x, x-y )|}{|x -y|^{n-\gamma-1}}|f_{j}(y)| \left(\int\limits_{|x|}^{\infty}\frac{dt}{t^{3}}\right)^{\frac{1}{2}}dy\\ &\leq C \int\limits_ {\mathbb{R}^{n}}\frac{|\Omega(x, x-y )|}{|x -y|^{n-\gamma-1}}|f_{j}(y)| \frac{1}{|x|}dy\\ &\leq C \int\limits_ {C_{j}}\frac{|\Omega(x, x-y )|}{|x -y|^{n-\gamma}}|f_{j}(y)|dy. \end{align*} So, we obtain that $$ |\mu_{\Omega,\gamma}f_{j}(x)| \leq C \int\limits_ {C_{j}}\frac{|\Omega(x, x-y )|}{|x -y|^{n-\gamma}}|f_{j}(y)|dy. $$ By the generalized Hölder's inequality assures that Hence, for any \(j\leq k-2\) and \(\Omega\in{L}^{\infty}(\mathbb{R}^{n})\times{L^{r}(S^{n-1})}\), we have Again applying the generalized Hölder's inequality, we have $$ \|f_{j}(y)\|_{L^{r'}(\mathbb{R}^{n})}\leq |B_{j}|^{-\frac{1}{r}}\|f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_j}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})}. $$ We can define a variable exponent \({\widetilde{p_{1}}(\cdot)}> 0\) by \(\frac{1}{r'}=\frac{1}{p_{1}(\cdot)}+\frac{1}{{\widetilde{p_{1}}(\cdot)}}\) due to \(1\leq r' < p^{-}\), we have Moreover, we can observe that \( \frac{1}{{\widetilde{p_{1}}(\cdot)}}=\frac{1}{r'}-\frac{1}{{p_{1}(\cdot)}}=\frac{1}{{p'_{1}(\cdot)}}-\frac{1}{r}\), applying Lemma 2.6, we have By (11)-(14), it follows that $$ |\mu_{\Omega,\gamma}f_{j}(x)| \leq C 2^{-k(n-\gamma)}2^{(k-j)\frac{n}{r}} \|f_{j}\|_{L^{p_{1}(\cdot)}}\|\chi_{B_{j}}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})}. $$ Thus, we obtain \begin{align*} &\left\| 2^{k\alpha(\cdot)}\sum\limits_{j=-\infty}^{k-2}|\mu_{\Omega,\gamma}f_{j}|\chi_{k}\right\|_{L^{p_{2}(\cdot)}}\\ &\leq C \sum\limits_{j=-\infty}^{k-2} 2^{-k(n-\gamma)}2^{(k-j)\frac{n}{r}}2^{(k-j)\alpha_{+}} \|2^{j\alpha(\cdot)}f_{j}\|_{L^{p_{1}(\cdot)}} \|\chi_{B_{j}}\|_{L^{{p_{1}'}(\cdot)}} \|\chi_{B_{k}}\|_{L^{p_{2}(\cdot)}}. \end{align*} Noting that, if \(\frac{1}{p_{1}(\cdot)}-\frac{1}{p_{2}(\cdot)}=\frac{\gamma}{n}\), then $$C_{1}|B|^{\frac{\gamma}{n}}\|\chi_{B}\|_{L^{p_{2}(\cdot)}}\leq \|\chi_{B}\|_{L^{p_{1}(\cdot)}} \leq C_{2}|B|^{\frac{\gamma}{n}}\|\chi_{B}\|_{L^{p_{2}(\cdot)}}$$ (see[22], p.370).\\ From this, and using Lemmas 2.7-2.8, we have Where \(\xi= (n\delta_{1}- \frac{n}{r}-\alpha_{+} )\), from condition (7) noting that \(\xi>0\). Now we consider two cases \(1< q_{1}< \infty\) and \(0< q_{1}\leq1\).\\ If \(0< q_{1}\leq1\), we have Case 2: If \(1< q_{1}< \infty\), applying Hölder's inequality, we obtain that Finally, we consider \(L_{3}\). Again by Lemma 2.1 we have We consider F. Let \( x \in C_{k}, j \geq k+2 \), then \( | x- y|\sim |y|\), we have \begin{align*} |\mu_{\Omega,\gamma}f_{j}(x)| &\leq \left(\int\limits_{0}^{|y|}\left|\int_{|x-y|\leq t}\frac{\Omega(x, x-y )}{|x -y|^{n-\gamma-1}}f_{j}(y)dy\right|^{2}\frac{dt}{t^{3}}\right)^{\frac{1}{2}}\\ &\quad+ \left(\int\limits_{|y|}^{\infty}\left|\int_{|x-y|\leq t}\frac{\Omega(x, x-y )}{|x -y|^{n-\gamma-1}}f_{j}(y)dy\right|^{2}\frac{dt}{t^{3}}\right)^{\frac{1}{2}}\\ &= I_{3}+ I_{4}. \end{align*} By (10), and the Minkowski inequality, we show that \begin{align*} I_{3}&\leq C \int\limits_ {\mathbb{R}^{n}}\frac{|\Omega(x, x-y )|}{|x -y|^{n-\gamma-1}}|f_{j}(y)| \left(\int\limits_{|x-y|}^{|y|}\frac{dt}{t^{3}}\right)^{\frac{1}{2}}dy\\ &\leq C \int\limits_{ \mathbb{R}^{n}}\frac{|\Omega(x, x-y )|}{|x -y|^{n-\gamma-1}}|f_{j}(y)| \left|\frac{1}{|x-y|^{2}}-\frac{1}{|y|^{2}}\right|^{\frac{1}{2}}dy\\ &\leq C \int\limits_{ \mathbb{R}^{n}}\frac{|\Omega(x, x-y )|}{|x -y|^{n-\gamma-1}}|f_{j}(y)| \frac{|x|^{\frac{1}{2}}}{|x-y|^{\frac{3}{2}}}dy\\ &\leq C {2^{(k-j)\frac{1}{2}}}\int\limits_{C_{j}}\frac{|\Omega(x, x-y )|}{|x -y|^{n-\gamma}}|f_{j}(y)| dy\\ &\leq C \int\limits_{ C_{j}}\frac{|\Omega(x, x-y )|}{|x -y|^{n-\gamma}}|f_{j}(y)| dy. \end{align*} Similarly, we estimate \(I_{4}\). By the Minkowski inequality and, we get \begin{align*} &I_{4}\leq C \int\limits_ {\mathbb{R}^{n}}\frac{|\Omega(x, x-y )|}{|x -y|^{n-\gamma-1}}|f_{j}(y)| \left(\int\limits_{|y|}^{\infty}\frac{dt}{t^{3}}\right)^{\frac{1}{2}}dy\\ &\leq C \int\limits_ {\mathbb{R}^{n}}\frac{|\Omega(x, x-y )|}{|x -y|^{n-\gamma-1}}|f_{j}(y)| \frac{1}{|y|}dy\\ &\leq C \int\limits_ {C_{j}}\frac{|\Omega(x, x-y )|}{|x -y|^{n-\gamma}}|f_{j}(y)|dy. \end{align*} So, we have $$ |\mu_{\Omega,\gamma}f_{j} (x)|\leq C \int\limits_ {C_{j}}\frac{|\Omega(x, x-y )|}{|x -y|^{n-\gamma}}|f_{j}(y)|dy . $$ Applying the generalized Hölder's inequality assures that Hence, for any \(j\geq k+2\) and \(\Omega\in{L}^{\infty}(\mathbb{R}^{n})\times{L^{r}(S^{n-1})}\), we can easy get The same way which was used in (13) and (14), we have Combining (20) with (21), we obtain that By (22) and again applying Lemmas 2.7-2.8, we show that Where \( \sigma=(n\delta_{2}-\gamma)\). Now also we have two cases \(1< q_{1}< \infty\) and \(0< q_{1}\leq1\). If \(0< q_{1}\leq1\), by (23) and \(\sigma +\alpha(0)>\sigma +\alpha_{-} \), we obtain that \begin{align*} F &\leq C \sup\limits_{k_{0}< 0,k_{0}\in \mathbb{Z}} 2^{-k_{0}\lambda {q_{1}}} \sum\limits_{k=-\infty}^{k_{0}} 2^{k\alpha(0)q_{1}}\sum\limits_{j=k+2}^{\infty}2^{(k-j)\sigma q_{1}} \|f_{j}\|^{q_{1}}_{L^{p_{1}(\cdot)}}\\ &\leq C \sup\limits_{k_{0}< 0,k_{0}\in \mathbb{Z}} 2^{-k_{0}\lambda {q_{1}}} \sum\limits_{k=-\infty}^{k_{0}} 2^{k\alpha(0)q_{1}}\sum\limits_{j=k+2}^{k_{0}-1}2^{(k-j)\sigma q_{1}} \|f_{j}\|^{q_{1}}_{L^{p_{1}(\cdot)}}\\ &~~~~+ \sup\limits_{k_{0}< 0,k_{0}\in \mathbb{Z}} 2^{-k_{0}\lambda {q_{1}}} \sum\limits_{k=-\infty}^{k_{0}} 2^{k\alpha(0)q_{1}}\sum\limits_{j=k_{0}}^{\infty}2^{(k-j)\sigma q_{1}} \|f_{j}\|^{q_{1}}_{L^{p_{1}(\cdot)}}\\ &= F_{1}+F_{2}. \end{align*} For \(F_{1} \), we have For \(F_{2} \), since \( \lambda-\sigma-\alpha(0)< \lambda-\sigma-\alpha_{-}\), we obtain that Case 2: If \(1< q_{1}< \infty\), we get \begin{align*} F &\leq C \sup\limits_{k_{0}< 0,k_{0}\in \mathbb{Z}} 2^{-k_{0}\lambda {q_{1}}} \sum\limits_{k=-\infty}^{k_{0}} 2^{k\alpha(0)q_{1}}\left(\sum\limits_{j=k+2}^{k_{0}}2^{(k-j)\sigma \|f_{j}\|_{L^{p_{1}(\cdot)}}}\right)^{q_{1}}\\ &+ \sup\limits_{k_{0}< 0,k_{0}\in \mathbb{Z}} 2^{-k_{0}\lambda {q_{1}}} \sum\limits_{k=-\infty}^{k_{0}} 2^{k\alpha(0)q_{1}}\left(\sum\limits_{j=k_{0}+1}^{\infty}2^{(k-j)\sigma \|f_{j}\|_{L^{p_{1}(\cdot)}}}\right)^{q_{1}}\\ &= F_{3} + F_{4}. \end{align*} For \(F_{3}\), using Hölder's inequality, we have For \(F_{4}\), we have We can omit the estimate of \(G \) since it is essentially similar to that of \(F\). Hence the proof of Theorem 3.1 is completed.

Theorem 3.2. Suppose that \(b\in Lip_{\beta}(\mathbb{R}^{n})\), \( m\in\mathbb{N}\), \( 0< \gamma< n,0< q\leq\infty,\) and \(\Omega\in{L}^{\infty}(\mathbb{R}^{n})\times{L^{r}(S^{n-1})} \) \((1< r \leq \infty) \). Let \(p_{1}(\cdot), p_{2}(\cdot) \in \mathfrak{B}(\mathbb{R}^{n})\), \(1\leq r' < (p_{1})_{-}\leq (p_{1})_{+} < \frac{n}{\gamma+m\beta}\), \(\frac{1}{p_{1}(x)} - \frac{1}{p_{2}(x)}= \frac{\gamma+m\beta}{n}\) and let \(\alpha(x)\in L^{\infty}(\mathbb{R}^{n})\) be log-Hölder continuous both at the origin and at infinity and satisfying (1.7). Then \([b^{m},\mu_{\Omega,\gamma}]\) is bounded from \({M\dot{K}_{q_{1},p_{1}(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})}\) to \({M\dot{K}_{q_{2},p_{2}(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})}\).

Proof. If \(f \in {M\dot{K}_{q_{1},p_{1}(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})}\), we applying \(\left[ \sum\limits_{j=1}^{\infty}a_{j}\right]^{q_{1}/q_{2}}\leq \sum\limits_{j=1}^{\infty}a_{j}^{q_{1}/q_{2}}\), \( a_{1} , a_{2} .... \geq 0.\) \begin{align*} &\left\|[b^{m},\mu_{\Omega , \gamma} ]f_{j}\right\|^{q_{1}}_{M\dot{K}_{q_{2},p_{2}(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})}\\&= \sup\limits_{k_{0}\in \mathbb{Z}} 2^{-k_{0}\lambda {q_{1}}} \left\{\sum_{k=-\infty}^{k_{0}} \left\|2^{k\alpha(\cdot)}[b^{m},\mu_{\Omega,\gamma}] (f)\chi_{k}\right\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}^{q_{2}}\right\}^{q_{1}/q_{2}}\\ &\leq \sup\limits_{k_{0}\in \mathbb{Z}} 2^{-k_{0}\lambda {q_{1}}} \sum\limits_{k=-\infty}^{k_{0}} \left\|2^{k\alpha(\cdot)}[b^{m},\mu_{\Omega,\gamma}] (f)\chi_{k}\right\|^{q_{1}}_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}. \end{align*} If we denote $$ f(x) = \sum_{j=-\infty}^{\infty}f(x)\chi_{j}= {\sum_{j=- \infty}^{ \infty}} f_{j}(x). $$ Then we have \begin{align*} &\left\|[b^{m},\mu_{\Omega,\gamma} ]f_{j}\right\|^{q_{1}}_{M\dot{K}_{q_{2},p_{2}(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})}\\ &\leq \sup\limits_{k_{0}\in \mathbb{Z}} 2^{-k_{0}\lambda {q_{1}}} \sum\limits_{k=-\infty}^{k_{0}} \left\|2^{k\alpha(\cdot)}\left(\sum\limits_{j=-\infty}^{\infty}|[b^{m},\mu_{\Omega,\gamma}] f_{j}|\right)\chi_{k}\right\|^{q_{1}}_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\\ &\leq \sup\limits_{k_{0}\in \mathbb{Z}} 2^{-k_{0}\lambda {q_{1}}} \sum\limits_{k=-\infty}^{k_{0}} \left\|2^{k\alpha(\cdot)}\left(\sum\limits_{j=-\infty}^{k-2}|[b^{m},\mu_{\Omega,\gamma}] f_{j}|\right)\chi_{k}\right\|^{q_{1}}_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\\ &\quad+ \sup\limits_{k_{0}\in \mathbb{Z}} 2^{-k_{0}\lambda {q_{1}}} \sum\limits_{k=-\infty}^{k_{0}} \left\|2^{k\alpha(\cdot)}\left(\sum\limits_{j=k-1}^{k+1}|[b^{m},\mu_{\Omega,\gamma}] f_{j}|\right)\chi_{k}\right\|^{q_{1}}_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\\ &\quad+ \sup\limits_{k_{0}\in \mathbb{Z}} 2^{-k_{0}\lambda {q_{1}}} \sum\limits_{k=-\infty}^{k_{0}} \left\|2^{k\alpha(\cdot)}\left(\sum\limits_{j=k+2}^{\infty}|[b^{m},\mu_{\Omega,\gamma}] f_{j}|\right)\chi_{k}\right\|^{q_{1}}_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\\ &=L_{21}+ L_{22}+L_{23}. \end{align*} First, we consider \(L_{22}\). Noting that \([ b^{m} ,\mu_{\Omega,\gamma}]\) is bounded on \(L^{p(\cdot)}(\mathbb{R}^{n})\) (Lemmas 2.5), as argued about \(L_{12}\) in the proof of Theorem 3.1, immediately get

Now we consider \(L_{21}\), we have \begin{align*} |[b^{m},\mu_{\Omega,\gamma}f(x)]| &\leq \left(\int\limits_{0}^{|x|}\left|\int_{|x-y|\leq t}\frac{\Omega(x, x-y )}{|x -y|^{n-\gamma-1}}[b(x) - b(y)]f_{j}(y)dy\right|^{2}\frac{dt}{t^{3}}\right)^{\frac{1}{2}}\\ &\quad+ \left(\int\limits_{|x|}^{\infty}\left|\int_{|x-y|\leq t}\frac{\Omega(x, x-y )}{|x -y|^{n-\gamma-1}}[b(x) - b(y)]f_{j}(y)dy\right|^{2}\frac{dt}{t^{3}}\right)^{\frac{1}{2}}\\ &= J_{1}+ J_{2}. \end{align*} Noting that \( x \in C_{k}\), \(j \leq k-2 \), then \( | x- y|\sim |x|\sim |2^{k}|\). By (10) and Minkowski inequality, we show that \begin{align*} J_{1}&\leq C \int_ {\mathbb{R}^{n}}\frac{|\Omega(x, x-y )|}{|x -y|^{n-\gamma-1}}|b(x)- b(y)|^{m}|f_{j}(y)| \left(\int\limits_{|x-y|}^{|x|}\frac{dt}{t^{3}}\right)^{\frac{1}{2}}dy\\ &\leq C \int_{ \mathbb{R}^{n}}\frac{|\Omega(x, x-y )|}{|x -y|^{n-\gamma-1}}|b(x)- b(y)|^{m}|f_{j}(y)| \left|\frac{1}{|x-y|^{2}}-\frac{1}{|x|^{2}}\right|^{\frac{1}{2}}dy\\ &\leq C \int_{ \mathbb{R}^{n}}\frac{|\Omega(x, x-y )|}{|x -y|^{n-\gamma-1}}|b(x)- b(y)|^{m}|f_{j}(y)| \frac{|y|^{\frac{1}{2}}}{|x-y|^{\frac{3}{2}}}dy\\ &\leq C 2^{(j-k)\frac{1}{2}} \int\limits_{ C_{j}}\frac{|\Omega(x, x-y )|}{|x -y|^{n-\gamma}}|b(x)- b(y)|^{m}|f_{j}(y)| dy\\ &\leq C \int\limits_{ C_{j}}\frac{|\Omega(x, x-y )|}{|x -y|^{n-\gamma}}|b(x)- b(y)|^{m}|f_{j}(y)| dy. \end{align*} Similarly, we estimate \(J_{1}\). By the Minkowski inequality, we have \begin{align*} J_{2}&\leq C \int_ {\mathbb{R}^{n}}\frac{|\Omega(x, x-y )|}{|x -y|^{n-\gamma-1}}|b(x)- b(y)|^{m}|f_{j}(y)| \left(\int\limits_{|x|}^{\infty}\frac{dt}{t^{3}}\right)^{\frac{1}{2}}dy\\ &\leq C \int\limits_ {C_{j}}\frac{|\Omega(x, x-y )|}{|x -y|^{n-\gamma-1}}|b(x)- b(y)|^{m}|f_{j}(y)| \frac{1}{|x|}dy\\ &\leq C \int\limits_{ C_{j}}\frac{|\Omega(x, x-y )|}{|x -y|^{n-\gamma}}|b(x)- b(y)|^{m}|f_{j}(y)| dy. \end{align*} So, we obtain that \begin{align*} |[b^{m},\mu_{\Omega,\gamma}f(x)]| &\leq C \int\limits_{ C_{j}}\frac{|\Omega(x, x-y )|}{|x -y|^{n-\gamma}}|b(x)- b(y)|^{m}|f_{j}(y)| dy\\ &\leq C 2^{-k(n-\gamma)}\int\limits_{ C_{j}}|\Omega(x, x-y )||b(x)- b(y)|^{m}|f_{j}(y)| dy\\ &\leq C 2^{-k(n-\gamma)}|b(x)- b_{j}|^{m}\int\limits_{ C_{j}}|\Omega(x, x-y )||f_{j}(y)| dy\\ &\quad+ 2^{-k(n-\gamma)}\int\limits_{ C_{j}}|\Omega(x, x-y )||b(y)- b_{j}|^{m}|f_{j}(y)| dy\\ &= A_{1} + A_{2} \end{align*} For \(A_{1}\), it is easy to check that $$A_{1}\leq C 2^{-k(n-\gamma)}2^{(k-j)\frac{n}{r}}|b(x)- b_{j}|^{m} \|f_{j}(y)\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\|\chi_{B_{j}}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})}. $$ Now we consider \(A_{2}\). By the generalized Hölder's inequality, we have $$A_{2}\leq C 2^{-k(n-\gamma)}\|\Omega(x, x-y )\|_{L^{r}(\mathbb{R}^{n})}\|f_{j}(y)(b(y)- b_{j})^{m}\|_{L^{r'}(\mathbb{R}^{n})} $$ Similar to (13), (14) and applying Lemma 2.9, we have \begin{align*} &\quad\|f_{j}(y)(b(y)- b_{j})^{m}\|_{L^{r'}(\mathbb{R}^{n})}\\ &\leq \|f_{j}\|_{L^{p_{1}(\cdot)}} \|(b(y)-b_{j})^{m}\chi_{B_{j}}\|_{L^{\widetilde{p_{1}}(\cdot)}}\\ &\leq C \|b\|^{m}_{Lip_{\beta}} \|f_{j}\|_{L^{p_{1}(\cdot)}} |B_{j}|^{\frac{m\beta}{n}} \|\chi_{B_{j}}\|_{L^{\widetilde{p_{1}}(\cdot)}}\\ &\leq C \|b\|^{m}_{Lip_{\beta}} |B_{j}|^{-\frac{1}{r}} \|f_{j}\|_{L^{p_{1}(\cdot)}}|B_{j}|^{\frac{m\beta}{n}} \|\chi_{B_{j}}\|_{L^{{p'_{1}}(\cdot)}}. \end{align*} Thus, we obtain $$A_{2}\leq C \|b\|^{m}_{Lip_{\beta}} 2^{-k(n-\gamma)} 2^{(k-j)\frac{n}{r}} \|f_{j}\|_{L^{p_{1}(\cdot)}}|B_{j}|^{\frac{m\beta}{n}} \|\chi_{B_{j}}\|_{L^{{p'_{1}}(\cdot)}}$$ From \(A_{1}, A_{2}\) and again applying Lemma 2.9, we have \begin{align*} &\quad\left\| 2^{k\alpha(\cdot)}\sum\limits_{j=-\infty}^{k-2}[b^{m},\mu_{\Omega,\gamma}]f_{j},\chi_{k}\right\|_{L^{p_{2}(\cdot)}}\\ &\leq C \sum\limits_{j=-\infty}^{k-2} 2^{-k(n-\gamma)} 2^{(k-j)\frac{n}{r}}2^{(k-j)\alpha_{+}} \|2^{j\alpha(\cdot)}f_{j}\|_{L^{p_{1}(\cdot)}}\\&~\times \left[\|(b(x)- b_{j})^{m}\chi_{k}\|_{L^{p_{2}(\cdot)}}\|\chi_{B_{j}}\|_{L^{p_{1}'(\cdot)}} +\|b\|^{m}_{Lip_{\beta}}|B_{j}|^{\frac{m\beta}{n}} \|\chi_{j}\|_{L^{{p_{1}'}(\cdot)}} \|\chi_{B_{k}}\|_{L^{p_{2}(\cdot)}}\right]\\ &\leq C \|b\|^{m}_{Lip_{\beta}}\sum\limits_{j=-\infty}^{k-2} 2^{-k(n-\gamma)} 2^{(k-j)\frac{n}{r}}2^{(k-j)\alpha_{+}} \|2^{j\alpha(\cdot)}f_{j}\|_{L^{p_{1}(\cdot)}}\\&~\times \left[|B_{k}|^{\frac{m\beta}{n}}\|\chi_{B_{j}}\|_{L^{p_{1}'(\cdot)}} \|\chi_{k}\|_{L^{{p_{2}}(\cdot)}} +|B_{j}|^{\frac{m\beta}{n}} \|\chi_{j}\|_{L^{{p_{1}'}(\cdot)}} \|\chi_{B_{k}}\|_{L^{p_{2}(\cdot)}}\right]. \end{align*} Noting that, if \(\frac{1}{p_{1}(\cdot)}-\frac{1}{p_{2}(\cdot)}=\frac{\gamma+m\beta}{n}\), then \(C_{1}|B|^{\frac{\gamma+m\beta}{n}}\|\chi_{B}\|_{L^{p_{2}(\cdot)}}\leq \|\chi_{B}\|_{L^{p_{1}(\cdot)}} \leq C_{2}|B|^{\frac{\gamma+m\beta}{n}}\|\chi_{B}\|_{L^{p_{2}(\cdot)}}\) (see[22],p.370).
From this, and using Lemmas 2.7-2.8, we have Furthermore, when \(\alpha_{+} < n\delta_{1}-\frac{n}{r}\), then the same arguments as \(L_{11}\) before, we can conclude that for all \(0< q< \infty\), Finally, we consider \(L_{23}\). Again by Lemma 2.1, we have \begin{align*} L_{23}&\approx max \left\{{\sup\limits_{k_{0}< 0,k_{0}\in \mathbb{Z}} 2^{-k_{0}\lambda {q_{1}}} \sum\limits_{k=-\infty}^{k_{0}} \left\|2^{k\alpha(0)}\left(\sum\limits_{j=k+2}^{\infty}|[b^{m},\mu_{\Omega ,\gamma}] f_{j}|\right)\chi_{k}\right\|^{q_{1}}_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}},\right.\\ &\left.\sup\limits_{k_{0}\geq 0,k_{0}\in \mathbb{Z}} \left[2^{-k_{0}\lambda {q_{1}}} \sum\limits_{k=-\infty}^{-1} \left\|2^{k\alpha(0)}\left(\sum\limits_{j=k+2}^{\infty}|[b^{m},\mu_{\Omega ,\gamma}] f_{j}|\right)\chi_{k}\right\|^{q_{1}}_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\right.\right.\\ &\left.\left. + 2^{-k_{0}\lambda {q_{1}}} \sum\limits_{k=0}^{k_{0}} \left\|2^{k\alpha_\infty}\left(\sum\limits_{j=k+2}^{\infty}|[b^{m},\mu_{\Omega ,\gamma}] f_{j}|\right)\chi_{k}\right\|^{q_{1}}_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\right]\right\}\\ &= max\{L,M\}. \end{align*} New we can choose \(L\), then we have \begin{align*} |[b^{m},\mu_{\Omega,\gamma}f(x)]| &\leq \left(\int\limits_{0}^{|y|}\left|\int_{|x-y|\leq t}\frac{\Omega(x, x-y )}{|x -y|^{n-\gamma-1}}[b(x) - b(y)]f_{j}(y)dy\right|^{2}\frac{dt}{t^{3}}\right)^{\frac{1}{2}}\\ &\quad+ \left(\int\limits_{|y|}^{\infty}\left|\int_{|x-y|\leq t}\frac{\Omega(x, x-y )}{|x -y|^{n-\gamma-1}}[b(x) - b(y)]f_{j}(y)dy\right|^{2}\frac{dt}{t^{3}}\right)^{\frac{1}{2}}\\ &= J_{3}+ J_{4}. \end{align*} Similar to the estimates of \( J_{1} , J_{2} \), such that \( x \in C_{k}, j \geq k+2 \), we have \begin{align*} |[b^{m},\mu_{\Omega,\gamma}f(x)]| &\leq C \int\limits_{ C_{j}}\frac{|\Omega(x, x-y )|}{|x -y|^{n-\gamma}}|b(x)- b(y)|^{m}|f_{j}(y)| dy\\ &\leq C 2^{-j(n-\gamma)}\int\limits_{ C_{j}}|\Omega(x, x-y )||b(x)- b(y)|^{m}|f_{j}(y)| dy\\ &\leq C 2^{-j(n-\gamma)}|b(x)- b_{j}|^{m}\int\limits_{ C_{j}}|\Omega(x, x-y )||f_{j}(y)| dy\\ &\quad+ 2^{-j(n-\gamma)}\int\limits_{ C_{j}}|\Omega(x, x-y )||b(y)- b_{j}|^{m}|f_{j}(y)| dy\\ &= A_{3} + A_{4}. \end{align*} For \(A_{3}\), it is easy to check that $$ A_{3}\leq C 2^{-j(n-\gamma)}|b(x)- b_{j}|^{m} \|f_{j}(y)\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\|\chi_{B_{j}}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})}. $$ By the same way which was used in \(A_{2}\), we obtain that $$ A_{4}\leq C \|b\|^{m}_{Lip_{\beta}} 2^{-j(n-\gamma)} \|f_{j}\|_{L^{p_{1}(\cdot)}}|B_{j}|^{\frac{m\beta}{n}} \|\chi_{j}\|_{L^{{p'_{1}}(\cdot)}}.$$ From \(A_{3}\) and \(A_{4}\), and also applying Lemmas 2.7-2.8, we show that Furthermore, when \(\alpha_{+}\leq\lambda- n\delta_{2}+\gamma\), the similar way to the estimate of \(L_{13}\) before, we can easily get for all \(0< q< \infty\), We can omit the estimate of \(M\) since it is essentially similar to that of \(L\). Hence the proof of Theorem \ref{3.2} is completed.

Remark If the variable exponent \(\alpha(\cdot)\) is constant, our results of this paper all hold.

Competing Interests

The authors declare that they have no competing interests.