Boundedness of Calderón-Zygmund operators and their commutator on Morrey-Herz Spaces with variable exponents

1. Introduction

Let \(K\) be a locally integrable function on \(\mathbb{R}^{n}\times\mathbb{R}^{n}\backslash \{(x,y): x=y\}\), then we say that \(K\) is a standard kernel if there exist \(\varepsilon > 0\) and \(C>0\), such that \begin{align*} |K(x,y)| &\leq C /|x-y|^{n}, x\neq y;\\ |K(x,y)-K(x,w)| &\leq C \frac{|y-w|^{\varepsilon}}{|x-y|^{n+\varepsilon}},|y-w| \leq \frac{1}{2} |x-y|;\\ |K(x,y)-K(z,y)| &\leq C \frac{|x-z|^{\varepsilon}}{|x-y|^{n+\varepsilon}},|x-z| \leq \frac{1}{2} |x-y|. \end{align*} We say that a linear operator \(T : \mathcal{S}(\mathbb{R}^{n})\longrightarrow \mathcal{S^{\prime}}(\mathbb{R}^{n})\) is a Calderón\(-\)Zygmund operator associated to a standard kernel \(K\) if

• 1. \(T\) can be extended to a bounded operator on \(L^{2}(\mathbb{R}^{n});\)
• 2. for all \(h\in L^{2}(\mathbb{R}^{n})\) with compact support and almost everywhere \(x\notin\) supp \( h\), \[Th(x)= \int_{\mathbb{R}^{n}} K(x,y)h(y)dy. \]

Now, suppose that \(b\in BOM(\mathbb{R}^{n})\) and \(T\) be a Calderón\(-\)Zygmund operators. The commutator \([b,T]\) generated by \(b\) is defined by In recent decades, the generalized Lebesgue spaces with variable exponent and the corresponding Sobolev spaces with variable exponent have attracted attention of researchers. Due to the fundamental paper [1] by Kováˇcik and Rákosník appeared in 1991, the theory of these spaces made progress rapidly and these studies have many applications in partial differential equations, fluid dynamics and image restoration [2,3,4,5]. One of the main problems on the theory of function spaces is the boundedness of the Hardy-Littlewood maximal operator on Lebesgue spaces with variable exponent. Many researchers [6,7,8,9] considered the question of sufficient conditions on the exponent function \(p(x)\) to obtained the boundedness of Hardy-Littlewood maximal operators.

Jouné proved that if \(T\) is a \({\varepsilon}\)-Calderón\(-\)Zygmund operator, then \(T\) is bounded on \(L^{p}(\mathbb{R}^{n})\) [10]. Coifman, Rochberg and Weiss proved that the commutator \([b,T]\) is bounded on \(L^{p}(\mathbb{R}^{n}) (1 < p < 1)\) [11]. In 1997, Lu [12] showed the commutator \([b,T\) on Herz-Type spaces. In 2006, Cruz-Uribe et al., [13] established the boundedness of some classical operators on variable \(L^{p}\) spaces by applying the theory of weighed norm inequalities and extrapolation.

The Morrey-Herz spaces have been playing a central role in harmonic analysis [14]. The boundedness of some operators and their corresponding characterization of these spaces with variable exponent \(p(x)\) were studied widely [15,16]. Recently, Morrey-Herz spaces \(MK_{q(\cdot),p(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})\) and \(M\dot{K}_{q(\cdot),p(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})\) with three variable exponents were studied by Wang and Tao [17].

2. Definition of function spaces with variable exponent

In this section we will recall the definition of Lebesgue spaces with variable exponents and the Morrey-Herz spaces with three variable exponents. Let \(\Omega\) be a measurable set in \(\mathbb{R}^{n}\) with \(|\Omega|> 0 \).

Definition 1.[11] Let \(p(\cdot): \Omega \rightarrow {[1,\infty)}\) be a measurable function, the Lebesgue space with variable exponent \(L^{p(\cdot)}(\Omega)\) is defined by \[L^{p(\cdot)}(\Omega)= \left\{{ h \mbox{is measurable} : \int_{\Omega}\left(\frac{|h(x)|}{\eta}\right)^{p(x)} dx < \infty} \mbox{for some constant } \eta > 0\right\}.\] The space \(L _{Loc}^{p(\cdot)} {(\Omega)}\) is defined by \(L_{Loc}^{p(\cdot)} {(\Omega)}= \{ \mbox {h is measurable} : h\in {L^{p(\cdot)} {(K)}}\) for all compact \(K\subset{\Omega}\)}. The Lebesgue spaces \(L^{p(\cdot)} {(\Omega)}\) is a Banach spaces with the norm defined by \[\|h\|_{L^{p(\cdot)}(\Omega)}= \inf\{\eta> 0 : \int_{\Omega}\left(\frac{|h(x)|}{\eta}\right)^{p(x)}dx \leq 1\},\] where \(p_{-}=\) ess \(\inf\{p(x): x \in \Omega\}, \) \( p_{+}=\) ess \(\sup \{p(x): x \in \Omega\} \). Then \(\mathcal{P}(\Omega)\) consists of all \(p(\cdot)\) satisfying \(p_{-} > 1\) and \(p_{+} < \infty\).

Let \(M\) be the Hardy-Littlewood maximal operator. We denote \(\mathcal{B}(\Omega)\) to be the set of all function \(p(\cdot)\in \mathcal{P}(\Omega)\) such that \(M\) is bounded on \(L^{p(\cdot)}(\Omega)\).

Let us turn to recall the definition of Herz spaces and Herz-Morrey spaces with variable exponents. We use the following notation;

Let \(B_{k}=\{ x\in\mathbb{R}^{n}:|x|\leq 2^{k}\}, C_{k}= B_{k}\backslash B_{k-1}, \chi_{k}= \chi_{C_{k}},k\in{\mathbb{Z}}.\)

Definition 2.[17] Let \(p(\cdot),q(\cdot)\in \mathcal{P}(\mathbb{R}^{n}),\alpha(\cdot): \mathbb{R}^{n}\longrightarrow\mathbb{R} \) with \( \alpha\in L^{\infty}(\mathbb{R}^{n})\) and \(0\leq \lambda < \infty.\) The nonhomogeneous Morrey-Herz space with variable exponent \(MK_{q(\cdot),p(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})\) and homogeneous Morrey-Herz space with variable exponents \(M\dot{K}_{q(\cdot),p(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})\) are defined by \[MK_{q(\cdot),p(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})= \left\{h\in {L_{\mathrm{loc}}^{p(\cdot)}}(\mathbb{R}^{n}\backslash\{0\}) : \|h\|_{{MK}_{q(\cdot),p(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})}< \infty\right \},\] and \[M\dot{K}_{q(\cdot),p(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})= \left\{h\in {L_{Loc}^{p(\cdot)}}(\mathbb{R}^{n}\backslash\{0\}) : \|h\|_{M\dot{K}_{q(\cdot),p(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})}< \infty \right\},\] respectively, where \begin{align*} \|h\|_{MK_{q(\cdot),p(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})} &= \inf\left\{ \eta> 0 : \sup\limits_{k_{0}\in z } 2^{-k_{0}\lambda} \sum\limits_{k=0}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}|h\chi_{k}|}{\eta}\right)^{q(\cdot)}\right\|_{L^{\frac{p(\cdot)}{q(\cdot)}}}\leq 1\right\},\\ \|h\|_{M\dot{K}_{q(\cdot),p(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})} &=\inf\left\{ \eta> 0 : \sup\limits_{k_{0}\in z} 2^{-k_{0}\lambda} \sum\limits_{k=-\infty}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}|h\chi_{k}|}{\eta}\right)^{q(\cdot)}\right\|_{L^{\frac{p(\cdot)}{q(\cdot)}}}\leq 1\right\}. \end{align*}

Remark 1.[17] Let \(v\in \mathbb{N},a_{v}\geq 0,1\leq p_{v} < \infty\). Then \(\sum\limits_{v=0}^{\infty} a_{v}\leq \left(\sum\limits_{v=0}^{\infty} a_{v} \right)^{p_{\ast}},\) where \( p_{\ast}= \left\{\begin{array}{ll} \min\limits_{v\in \mathbb{N} }p_{v}, \sum\limits_{v=0}^{\infty} a_{v}\leq 1,\\ \max\limits_{v\in \mathbb{N} }p_{v}, \sum\limits_{v=0}^{\infty} a_{v}>1. \end{array}\right.\)

Definition 3.[18] For all \(0< \beta \leq 1,\) the Lipschitz space \(Lip_{\beta}(\mathbb{R}^{n})\) is defined by \[Lip_{\beta}(\mathbb{R}^{n})=\left\{h:\|h\|_{Lip_{\beta}(\mathbb{R}^{n})}= \sup\limits_{x,y\in \mathbb{R}^{n};x\neq y}\frac{|h(x)-h(y)|}{|x-y|^{\beta}}< \infty\right\}.\]

3. Properties and lemmas of variable exponent

Proposition 1.[19] If \(p(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\), then \begin{align*} |p(x) - p(y)|\leq \frac{ -C}{Log( |x - y|)},& \;\;\text{if}\;\;| x - y| \leq 1/ 2\,,\\ | p(x) - p(y)|\leq \frac{ C}{Log( e +|x|)}, & \;\;\text{if}\;\; |y|\geq|x|. \end{align*}

Lemma 1.[1] Let \(p(\cdot)\in\mathcal{P}(\mathbb{R}^{n})\). If \(h\in L^{p(\cdot)}\) and \(g\in L^{p'(\cdot)}\), then \(hg\) is integrable on \(\mathbb{R}^{n}\) and \[\int_{\mathbb{R}^{n}}|h(x) g(x)| dx \leq C_{p}\|h\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\|g\|_{L^{p'(\cdot)}(\mathbb{R}^{n})}\,,\] where \(C_{p}=1+\frac{1}{p_{-}}-\frac{1}{p_{+}}\).

Lemma 2.[1] Suppose that \(p(\cdot),p_{1}(\cdot),p_{2}(\cdot)\in\mathcal{P}(\mathbb{R}^{n})\) and for any \(h\in L^{p_{1}(\cdot)}(\mathbb{R}^{n}),\;\;g\in L^{p_{2}(\cdot)}(\mathbb{R}^{n})\), when \(\frac{1}{p(\cdot)}=\frac{1}{p_{2}(\cdot)}+\frac{1}{p_{1}(\cdot)}\), we get \[\|h(x)g(x)\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\leq C \|h\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \|g\|_{L^{p_{2}}(\mathbb{R}^{n})}\,,\] where \(C_{p_{1},p_{2}}=[1+\frac{1}{p_{1-}}-\frac{1}{p_{1+}}]^{\frac{1}{p_{-}}}\).

Lemma 3.[20] Let \(b\in BMO(\mathbb{R}^{n})\) and \(i,j\in\mathbb{Z}\) with \(i< j\), then

• 1. \(C^{-1}\|b\|_{BMO(\mathbb{R}^{n})}\leq\sup\limits_{B}\frac{1}{\|\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}} \|(b-b_{B})\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})} \leq C\|b\|_{BMO(\mathbb{R}^{n})};\)
• 2. \(\|(b-b_{B_{i}})\chi_{B_{j}}\|_{L^{q(\cdot)}(\mathbb{R}^{n})}\leq C( j-i)\|b\|_{BMO(\mathbb{R}^{n})}\|\chi_{B_{j}}\|_{L^{q(\cdot)}(\mathbb{R}^{n})}.\)

Lemma 4.[21,22] Let \(p_{u}(\cdot) \in \mathfrak{B}(\mathbb{R}^{n})(u=1,2),\) then there exist constants \(0< \delta_{u1},\delta_{u2}< 1\) and \(C > 0\) such that for all balls \(B\subset\mathbb{R}^{n}\) and all measurable subset \(R\subset B,\) we have \[\frac{\|\chi_{B}\|_{L^{p_{u}(\cdot)}(\mathbb{R}^{n})}}{\|\chi_{R}\|_{L^{p_{u}(\cdot)}(\mathbb{R}^{n})}}\leq C \frac{|B|}{|R|}, \frac{\|\chi_{R}\|_{L^{p_{u}(\cdot)}(\mathbb{R}^{n})}}{\|\chi_{B}\|_{L^{p_{u}(\cdot)}(\mathbb{R}^{n})}}\leq C\left(\frac{|R|}{|B|}\right)^{\delta_{u2}}, \frac{\|\chi_{R}\|_{L^{p_{u}^{\prime}(\cdot)}(\mathbb{R}^{n})}}{\|\chi_{B}\|_{L^{p_{u}^{\prime}(\cdot)}(\mathbb{R}^{n})}}\leq C\left(\frac{|R|}{|B|}\right)^{\delta_{u1}}.\]

Lemma 5.[11] If \(p(\cdot) \in \mathfrak{B}(\mathbb{R}^{n}),\) there exist a 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 6.[11] Suppose \(p(\cdot),q(\cdot) \in\mathcal{P}(\mathbb{R}^{n}).\) If \(h\in L^{p(\cdot)q(\cdot)},\) then \[\min \left( \|h\|_{L^{p(\cdot)q(\cdot)}}^{q_{+}}, \|h\|_{L^{p(\cdot)q(\cdot)}}^{q_{-}} \right)\leq\||h|^{q(\cdot)}\|_{L^{p(\cdot)}}\leq\max \left( \|h\|_{L^{p(\cdot)q(\cdot)}}^{q_{+}}, \|h\|_{L^{p(\cdot)q(\cdot)}}^{q_{-}} \right) .\]

Proposition 2.[11] Let \(I_{\beta} \) be a fractional integrals operator \(p_{1}(\cdot),p_{2}(\cdot) \in \mathcal{B}(\mathbb{R}^{n})\) and \(0 < \beta< n/(p_{1})_{+}\). If \(\frac{1}{p_{1}(x)}-\frac{1}{p_{2}(x)}=\frac{\beta}{n}\), then we have \[\|I_{\beta}h\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \leq C \|h\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})},\] for all \(h\in L^{p_{1}(\cdot)}.\)

Lemma 7.[11] Suppose that \([b,T]\) as defined in (1) and \(p_{1}(\cdot),p_{2}(\cdot) \in \mathcal{B}(\mathbb{R}^{n})\). If \(b\in Lip_{\beta}(\mathbb{R}^{n})\) \((0< \beta< n/(p_{1})_{+})\) and \(\frac{1}{p_{1}(x)}-\frac{1}{p_{2}(x)}=\frac{\beta}{n}\), then \([b,T]\) is bounded from \(L^{p_{2}(\cdot)}(\mathbb{R}^{n})\) in to \(L^{p_{1}(\cdot)}(\mathbb{R}^{n})\).

Proof. Set \(b\in Lip_{\beta}(\mathbb{R}^{n})(0< \beta< 1)\), then \begin{align*} |[b,T](h)(x)|&\leq\int_{\mathbb{R}^{n}}|(b(x)-b(y))K(x,y)h(y)|dy\\&\leq\int_{\mathbb{R}^{n}}|(b(x)-b(y))\frac{C}{|x-y|^{n}}h(y)|dy\\ & \leq C \|b\|_{Lip_{\beta}(\mathbb{R}^{n})}\int_{\mathbb{R}^{n}}\frac{|h(y)|}{|x-y|^{n-\beta}}dy. \end{align*} Notice that \(0< \beta < n/(p_{1})_{+}\) so by applying Proposition 2, therefore \[\|[b,T](h)\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\leq C \|b\|_{Lip_{\beta}(\mathbb{R}^{n})} \|I_{\beta}(|h|)\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \leq C \|b\|_{Lip_{\beta}(\mathbb{R}^{n})}\|h\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}.\]

4. Main result and proof

Theorem 1. Suppose that \(p(\cdot)\in\mathcal{B}(\mathbb{R}^{n}),\;\;q_{1}(\cdot),q_{2}(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\) with \((q_{2})_{-}\geq(q_{1})_{+}\). If \(\lambda_{1}(q_{2})_{+}=\lambda_{2}(q_{1})_{-},\;\;\lambda_{1}/(q_{1})_{-}-n\delta_{12}< \alpha_{+}< \lambda_{1}/(q_{1})_{-}+n\delta_{11}\) with \(\delta_{11},\delta_{12}\) as in Lemma 4, then the operator \(T\) is bounded from \( MK^{\alpha_{+},\lambda_{1}}_{q_{1}(\cdot),p(\cdot)}(\mathbb{R}^{n})\) to \( MK^{\alpha(\cdot),\lambda_{2}}_{q_{2}(\cdot),p(\cdot)}(\mathbb{R}^{n})\).

Proof. Let \(h\in MK^{\alpha_{+},\lambda_{1}}_{q_{1}(\cdot),p(\cdot)}(\mathbb{R}^{n}).\) Write \[h(x)=\sum\limits_{j=0}^{\infty}h(x)\chi_{j}(x)\triangleq\sum\limits_{j=0}^{\infty}h_{j}(x).\] By the Definition 2, we get \begin{align*} \|T(h)\|_{MK^{\alpha(\cdot),\lambda_{2}}_{q_{2}(\cdot),p(\cdot)}(\mathbb{R}^{n})}=\inf\left\{ \eta> 0 : \sup\limits_{k_{0}\in z } 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}|T(h)\chi_{k}|}{\eta}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p(\cdot)}{q_{2}(\cdot)}}}\leq 1\right\}. \end{align*} For any \(k_{0}\in \mathbb{Z}\), we have \begin{align*} &2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}}\left\|\left(\frac{2^{k\alpha(\cdot)}|T(h)\chi_{k}|}{\eta}\right)^{q_{2}(\cdot)} \right\|_{L^{\frac{p(\cdot)}{q_{2}(\cdot)}}}\leq 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}\left|\sum\limits_{j=0}^{\infty}T(h_{j})\chi_{k}\right|}{\sum\limits_{i=1}^{3}\eta_{1i}} \right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p(\cdot)}{q_{2}(\cdot)}}}\\ &\leq 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}\left|\sum\limits_{j=0}^{k-2}T(h_{j})\chi_{k}\right|}{\eta_{11}} \right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p(\cdot)}{q_{2}(\cdot)}}}+2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}\left|\sum\limits_{j=k-1}^{k+1}T(h_{j})\chi_{k}\right|}{\eta_{12}} \right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p(\cdot)}{q_{2}(\cdot)}}}\\ &+ 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}\left|\sum\limits_{j=k+2}^{\infty}T(h_{j})\chi_{k}\right|}{\eta_{13}} \right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p(\cdot)}{q_{2}(\cdot)}}}. \end{align*} Let \begin{align*} \eta_{11}=\left\|\sum\limits_{j=0}^{k-2}T(h_{j})\right\|_{MK^{\alpha(\cdot),\lambda_{2}}_{q_{2}(\cdot),p(\cdot)}(\mathbb{R}^{n})}=\inf\left\{ \eta> 0 : \sup\limits_{k_{0}\in z } 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}\left|\sum\limits_{j=0}^{k-2}T(h_{j})\chi_{k}\right|}{\eta}\right)^{q_{2}(\cdot)} \right\|_{L^{\frac{p(\cdot)}{q_{2}(\cdot)}}}\leq 1\right\} \end{align*} \begin{align*} \eta_{12}&=\left\|\sum\limits_{j=k-1}^{k+1}T(h_{j})\right\|_{MK^{\alpha(\cdot),\lambda_{2}}_{q_{2}(\cdot),p(\cdot)}(\mathbb{R}^{n})}=\inf\left\{ \eta> 0 : \sup\limits_{k_{0}\in z } 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}\left|\sum\limits_{j=k-1}^{k+1}T(h_{j})\chi_{k}\right|}{\eta}\right)^{q_{2}(\cdot)} \right\|_{L^{\frac{p(\cdot)}{q_{2}(\cdot)}}}\leq 1\right\}\\ \eta_{13}&=\left\|\sum\limits_{j=k+2}^{\infty}T(h_{j})\right\|_{MK^{\alpha(\cdot),\lambda_{2}}_{q_{2}(\cdot),p(\cdot)}(\mathbb{R}^{n})}=\inf\left\{ \eta> 0 : \sup\limits_{k_{0}\in z } 2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}\left|\sum\limits_{j=k+2}^{\infty}T(h_{j})\chi_{k}\right|}{\eta}\right)^{q_{2}(\cdot)} \right\|_{L^{\frac{p(\cdot)}{q_{2}(\cdot)}}}\leq 1\right\} \end{align*} and \[\eta=\sum_{i=1}^{3}\eta_{1i}.\] Thus, we have \[2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}}\left\|\left(\frac{2^{k\alpha(\cdot)}|T(h)\chi_{k}|}{\eta}\right)^{q_{2}(\cdot)} \right\|_{L^{\frac{p(\cdot)}{q_{2}(\cdot)}}}\leq C.\] This implies that

Hence, it suffices to prove \[\eta_{11},\eta_{12} , \eta_{13} \leq C \eta_{10}\leq C \|h\|_{MK^{\alpha(\cdot),\lambda_{1}}_{q_{1}(\cdot),p(\cdot)}(\mathbb{R}^{n})}.\]

Denote \(\eta_{10}\leq C \|h\|_{MK^{\alpha(\cdot),\lambda_{1}}_{q_{1}(\cdot),p(\cdot)}(\mathbb{R}^{n})}\)

Step 1. We first estimate \(\eta_{12}\). By Lemma 6 and the \(T\)-boundedness in \(L^{p(\cdot)}\) (see [10]), we conclude that where \[ {(q^{1}_{2})k}= \left\{\begin{array}{ll} (q_{2})_{-}, \left\| \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k-1}^{k+1} T(h_{j}) \chi_{k}|}{\eta_{10}}\right\|_{L^{{p(\cdot)}}} \leq 1, \\ (q_{2})_{+}, \left\| \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k-1}^{k+1} T(h_{j}) \chi_{k}|}{\eta_{10}}\right\|_{L^{{p(\cdot)}}} >1. \end{array}\right. \] By applying Lemma 6 in (3) and assuming that \(\lambda_{1}(q_{2})_{+}=\lambda_{2}(q_{1})_{-},\) we have \begin{align*} &2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k-1}^{k+1}T(h_{j})\chi_{k}|}{\eta_{10}}\right)^{q_{2}(\cdot)} \right\|_{L^{\frac{p(\cdot)}{q_{2}(\cdot)}}}\leq 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}}\left(\sum\limits_{j=k-1}^{k+1} \left\| \frac{2^{j\alpha_{+}}|h_{j}|}{\eta_{10}}\right\|_{L^{p(\cdot)}}\right)^{(q_{2}^{1})_{k}}\\ &\leq 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}}\left( \left\| \frac{2^{k\alpha_{+}}|h\chi_{k}|}{\eta_{10}}\right\|_{L^{p(\cdot)}}\right)^{(q_{2}^{1})_{k}}\leq 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}}\left\|\left( \frac{2^{k\alpha_{+}}|h\chi_{k}|}{\eta_{10}}\right)^{q_{1}(\cdot)}\right\|_{L^{\frac{p(\cdot)}{q_{1}(\cdot)}}}^{\frac{(q_{2}^{1})_{k}}{(q_{1}^{1})_{k}}}\\ &\leq \sum\limits_{k=0}^{k_{0}}\left\{2^{-k_{0}\lambda_{1}}\left\|\left( \frac{2^{k\alpha_{+}}|h\chi_{k}|}{\eta_{10}}\right)^{q_{1}(\cdot)}\right\|_{L^{\frac{p(\cdot)}{q_{1}(\cdot)}}}\right\}^{\frac{(q_{2}^{1})_{k}}{(q_{1}^{1})_{k}}} \end{align*} where \[ {(q^{1}_{1})_{k}}= \left\{\begin{array}{ll} (q_{1})_{-}, \left\| \frac{2^{k\alpha_{+}}|h\chi_{k}|}{\eta_{10}}\right\|_{L^{{p(\cdot)}}} \leq 1, \\ (q_{1})_{+}, \left\|\frac{2^{k\alpha_{+}}|h\chi_{k}|}{\eta_{10}}\right\|_{L^{{p(\cdot)}}} >1. \end{array}\right.\] Since \(h\in MK^{\alpha_{+},\lambda_{1}}_{q_{1}(\cdot),p(\cdot)}(\mathbb{R}^{n}),\) it is easy to see that \[2^{-k_{0}\lambda_{1}}\left\|\left( \frac{2^{k\alpha_{+}}|h\chi_{k}|}{\eta_{10}}\right)^{q_{1}(\cdot)}\right\|_{L^{\frac{p(\cdot)}{q_{1}(\cdot)}}}\leq1.\] From above, with \((q_{1})_{+}\leq(q_{2})_{-}\), we get the following inequality \begin{align*} 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k-1}^{k+1}T(h_{j})\chi_{k}|}{\eta_{10}}\right)^{q_{2}(\cdot)} \right\|_{L^{\frac{p(\cdot)}{q_{2}(\cdot)}}} &\leq C \sum\limits_{k=0}^{k_{0}} 2^{-k_{0}\lambda_{1}}\left\|\left( \frac{2^{k\alpha_{+}}|h\chi_{k}|}{\eta_{10}}\right)^{q_{1}(\cdot)}\right\|_{L^{\frac{p(\cdot)}{q_{1}(\cdot)}}}\leq C. \end{align*} These imply that \[ \eta_{12} \leq C \eta_{10}\leq C \|h\|_{MK^{\alpha(\cdot),\lambda_{1}}_{q_{1}(\cdot),p(\cdot)}(\mathbb{R}^{n})}.\] Step 2. Let us turn to estimate \(\eta_{12}\). For each \(k\in \mathbb{Z},j \leq k-2\) and a.e. \(x\in R_{k},\) applying the generalized Hölder inequality, we have \begin{align*} |Th_{j}(x)|&\leq \int_{R_{k-2}}|K(x,y)||h_{j}(y)|dy\leq C 2^{-kn}\int_{R_{j}}|h_{j}(y)|dy \leq C 2^{-kn}\|h\|_{L^{1}(\mathbb{R}^{n})}. \end{align*} By Lemma 6, we have \begin{align*} &2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=0}^{k-2}T(h_{j})\chi_{k}|}{\eta_{10}}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p(\cdot)}{q_{2}(\cdot)}}} \leq 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}} \left\| \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=0}^{k-2}T(h_{j})\chi_{k}|}{\eta_{10}}\right\|^{(q^{2}_{2})_{k}}_{L^{p(\cdot)}}\\ & \leq C 2^{-k_{\circ}\lambda_{2}} \sum\limits_{k=0}^{k_{\circ}} \left\| \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=0}^{k-2}2^{-kn}\|h_{j}\|_{L^{1}(\mathbb{R}^{n})}\chi_{k}|}{\eta_{10}} \right\|^{(q^{2}_{2})_{k}}_{L^{p(\cdot)}} \leq C 2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}}\left\{2^{k\alpha_{+}} \sum\limits_{j=0}^{k-2}2^{-kn}\left\|\frac{h_{j}}{\eta_{10}}\right\|_{L^{1}(\mathbb{R}^{n})} \|\chi_{k}\|_{L^{p(\cdot)}}\right\}^{(q^{2}_{2})_{k}}, \end{align*} where \[ {(q^{2}_{2})_{k}}= \left\{\begin{array}{ll} (q_{2})_{-}, \left\| \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=0}^{k-2} T(h_{j}) \chi_{k}|}{\eta_{10}}\right\|_{L^{{p(\cdot)}}} \leq 1, \\ (q_{2})_{+}, \left\| \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=0}^{k-2} T(h_{j}) \chi_{k}|}{\eta_{10}}\right\|_{L^{{p(\cdot)}}} >1. \end{array}\right. \] By Lemmas 4 and 5, we have Applying Lemma 6 on (4), we obtain \begin{align*} &2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=0}^{k-2}T(h_{j})\chi_{k}|}{\eta_{10}}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p(\cdot)}{q_{2}(\cdot)}}} \leq C 2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}}\left\{ \sum\limits_{j=0}^{k-2}2^{(k-j)(\alpha_{+}-n\delta_{11})}\left\|\left(\frac{2^{j\alpha_{+}}h\chi_{j}}{\eta_{10}}\right)^{q_{1}(\cdot)} \right\|^{\frac{1}{(q^{2}_{1})_{j}}}_{L^{\frac{p(\cdot)}{q_{1}(\cdot)}}} \right\}^{(q^{2}_{2})_{k}}\\ &\leq C 2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}}2^{(k-k)\lambda_{2}}\left\{ \sum\limits_{j=0}^{k-2} 2^{(k-j)(\alpha_{+}-n\delta_{11})} \times\left(2^{j\lambda_{1}}2^{-j\lambda_{1}}\sum\limits_{\ell=0}^{j} \left\|\left(\frac{2^{\ell\alpha_{+}}h\chi_{j}}{\eta_{10}}\right)^{q_{1}(\cdot)} \right\|_{L^{\frac{p(\cdot)}{q_{1}(\cdot)}}}\right)^{\frac{1}{(q^{2}_{1})_{j}}}\right\}^{(q^{2}_{2})_{k}}\\ &\leq C \sum\limits_{k=0}^{k_{0}}2^{(k-k_{0})\lambda_{2}}\left\{ \sum\limits_{j=0}^{k-2} 2^{(k-j)(\alpha_{+}-n\delta_{11}-\lambda_{1}/(q_{1})_{-})}\right.\times\left.\left(2^{-j\lambda_{1}}\sum\limits_{\ell=0}^{j} \left\|\left(\frac{2^{\ell\alpha_{+}}h\chi_{j}}{\eta_{10}}\right)^{q_{1}(\cdot)} \right\|_{L^{\frac{p(\cdot)}{q_{1}(\cdot)}}}\right)^{\frac{1}{(q^{2}_{1})_{j}}}\right\}^{(q^{2}_{2})_{k}}, \end{align*} where \[{(q^{2}_{1})_{j}}= \left\{\begin{array}{ll} (q_{1})_{-}, \left\| \frac{2^{j\alpha_{+}}|h\chi_{j}|}{\eta_{10}}\right\|_{L^{{p(\cdot)}}} \leq 1, \\ (q_{1})_{+}, \left\|\frac{2^{j\alpha_{+}}|h\chi_{j}|}{\eta_{10}}\right\|_{L^{{p(\cdot)}}} >1. \end{array}\right.\] Noting that \(h\in MK^{\alpha_{+},\lambda_{1}}_{q_{1}(\cdot),p(\cdot)}(\mathbb{R}^{n})\) and \(\alpha_{+}< n\delta_{11}+\lambda_{1}/(q_{1})_{-},\) so we get \begin{align*} 2^{-k_{\circ}\lambda_{2}} \sum\limits_{k=0}^{k_{\circ}} \left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=0}^{k-2}T(h_{j})\chi_{k}|}{\eta_{10}}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p(\cdot)}{q_{2}(\cdot)}}} &\leq C \sum\limits_{k=0}^{k_{\circ}}2^{(k-k_{\circ})\lambda_{2}}\left( \sum\limits_{j=0}^{k-2} 2^{(k-j)(\alpha_{+}-n\delta_{11}-\lambda_{1}/(q_{1})_{-})}\right)^{(q^{2}_{2})_{k}}\leq C. \end{align*} This implies that \[ \eta_{12} \leq C \eta_{10}\leq C \|h\|_{MK^{\alpha(\cdot),\lambda_{1}}_{q_{1}(\cdot),p(\cdot)}(\mathbb{R}^{n})}. \] Step 3. Finally, we consider \( \eta_{13}\). For each \(j \geq k+2\) and \(x\in R_{k},y\in R_{j}\). By the similar argument in Step 2, we obtain that \[ |Th_{j}(x)| \leq C 2^{-jn}\|h\|_{L^{1}(\mathbb{R}^{n})}, \] and \begin{align*} &2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k+2}^{\infty}T(h_{j})\chi_{k}|}{\eta_{10}}\right)^{q_{2}(\cdot)} \right\|_{L^{\frac{p(\cdot)}{q_{2}(\cdot)}}}\leq 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}} \left\| \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k+2}^{\infty}T(h_{j})\chi_{k}|}{\eta_{10}}\right\|^{(q^{3}_{2})_{k}}_{L^{p(\cdot)}}\\ & \leq C 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}} \left\| \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k+2}^{\infty}2^{-jn}\|h_{j}\|_{L^{1}(\mathbb{R}^{n})}\chi_{k}|}{\eta_{10}} \right\|^{(q^{3}_{2})_{k}}_{L^{p(\cdot)}} \leq C 2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}}\left\{2^{k\alpha_{+}} \sum\limits_{j=k+2}^{\infty}2^{-jn}\left\|\frac{h_{j}}{\eta_{10}}\right\|_{L^{1}(\mathbb{R}^{n})} \|\chi_{k}\|_{L^{p(\cdot)}}\right\}^{(q^{3}_{2})_{k}}, \end{align*} where \[ {(q^{3}_{2})k}= \left\{\begin{array}{ll} (q_{2})_{-}, \left\| \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k+2}^{\infty} T(h_{j}) \chi_{k}|}{\eta_{10}}\right\|_{L^{{p(\cdot)}}} \leq 1, \\ (q_{2})_{+}, \left\| \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k+2}^{\infty} T(h_{j}) \chi_{k}|}{\eta_{10}}\right\|_{L^{{p(\cdot)}}} >1. \end{array}\right. \] So, by Lemmas 4, 5 and 6, we have \begin{align*} &2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k+2}^{\infty}T(h_{j})\chi_{k}|}{\eta_{10}}\right)^{q_{2}(\cdot)} \right\|_{L^{\frac{p(\cdot)}{q_{2}(\cdot)}}}\\ & \leq C 2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}}\left\{2^{k\alpha_{+}} \sum\limits_{j=k+2}^{\infty}2^{-jn} \times\left\|\frac{h_{j}}{\eta_{10}}\right\|_{L^{p(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{j}}\|_{L^{p^{\prime}(\cdot)}}\|\chi_{k}\|_{L^{p(\cdot)}}\right\}^{(q^{3}_{2})_{k}}\\ & \leq C 2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}}\left\{2^{k\alpha_{+}} \sum\limits_{j=k+2}^{\infty}2^{-jn} \times\left\|\frac{h_{j}}{\eta_{10}}\right\|_{L^{p(\cdot)}(\mathbb{R}^{n})} \frac{\|\chi_{k}\|_{L^{p(\cdot)}}}{\|\chi_{B_{j}}\|_{L^{p(\cdot)}}}|B_{j}|\right\}^{(q^{3}_{2})_{k}}\\ & \leq C 2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}}\left\{2^{k\alpha_{+}} \sum\limits_{j=k+2}^{\infty}2^{(k-j)n\delta_{12}}2^{-j\alpha_{+}}\left\|\frac{2^{j\alpha_{+}}h\chi_{j}}{\eta_{10}} \right\|_{L^{p(\cdot)}(\mathbb{R}^{n})} \right\}^{(q^{3}_{2})_{k}}\\ & \leq C 2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}}\left\{ \sum\limits_{j=k+2}^{\infty}2^{(k-j)(\alpha_{+}+n\delta_{12})}\left\|\left(\frac{2^{j\alpha_{+}}h\chi_{j}}{\eta_{10}}\right)^{q_{1}(\cdot)} \right\|^{\frac{1}{(q^{3}_{1})_{j}}}_{L^{\frac{p(\cdot)}{q_{1}(\cdot)}}}\right\}^{(q^{3}_{2})_{k}}, \end{align*} where \[{(q^{3}_{1})_{j}}= \left\{\begin{array}{ll} (q_{1})_{-}, \left\| \frac{2^{j\alpha_{+}}|h\chi_{j}|}{\eta_{10}}\right\|_{L^{{p(\cdot)}}} \leq 1, \\ (q_{1})_{+}, \left\|\frac{2^{j\alpha_{+}}|h\chi_{j}|}{\eta_{10}}\right\|_{L^{{p(\cdot)}}} >1. \end{array}\right.\] Hence, \(h\in MK^{\alpha_{+},\lambda_{1}}_{q_{1}(\cdot),p(\cdot)}(\mathbb{R}^{n})\) and \(-n\delta_{12}+\lambda_{1}/(q_{1})_{-}< \alpha_{+} ,\) so we get \begin{align*} &2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k+2}^{\infty}T(h_{j})\chi_{k}|}{\eta_{10}}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p(\cdot)}{q_{2}(\cdot)}}} \leq C 2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}}\left\{ \sum\limits_{j=k+2}^{\infty}2^{(k-j)(\alpha_{+}+n\delta_{12})}\left\|\left(\frac{2^{j\alpha_{+}}h\chi_{j}}{\eta_{10}}\right)^{q_{1}(\cdot)} \right\|^{\frac{1}{(q^{3}_{1})_{j}}}_{L^{\frac{p(\cdot)}{q_{1}(\cdot)}}}\right\}^{(q^{3}_{2})_{k}}\\ &\leq C 2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}}2^{-k_{0}\lambda_{2}}2^{(k-k)\lambda_{2}} \left\{ \sum\limits_{j=k+2}^{\infty}2^{(k-j)(\alpha_{+}+n\delta_{12})} \times\left(2^{j\lambda_{1}}2^{-j\lambda_{1}}\sum\limits_{\ell=0}^{j} \left\|\left(\frac{2^{\ell\alpha_{+}}h\chi_{j}}{\eta_{10}}\right)^{q_{1}(\cdot)} \right\|_{L^{\frac{p(\cdot)}{q_{1}(\cdot)}}}\right)^{\frac{1}{(q^{3}_{1})_{j}}}\right\}^{(q^{3}_{2})_{k}}\\ &\leq C \sum\limits_{k=0}^{k_{0}}2^{(k-k_{0})\lambda_{2}} \left\{\sum\limits_{j=k+2}^{\infty}2^{(k-j)(\alpha_{+}+n\delta_{12}-\lambda_{1}/(q_{1})_{-})} \times\left(2^{-j\lambda_{1}}\sum\limits_{\ell=0}^{j} \left\|\left(\frac{2^{\ell\alpha_{+}}h\chi_{j}}{\eta_{10}}\right)^{q_{1}(\cdot)} \right\|_{L^{\frac{p(\cdot)}{q_{1}(\cdot)}}}\right)^{\frac{1}{(q^{3}_{1})_{j}}}\right\}^{(q^{3}_{2})_{k}}\\ &\leq C \sum\limits_{k=0}^{k_{0}}2^{(k-k_{0})\lambda_{2}}\left( \sum\limits_{j=k+2}^{\infty} 2^{(k-j)(\alpha_{+}+n\delta_{12}-\lambda_{1}/(q_{1})_{-})}\right)^{(q^{3}_{2})_{k}}\leq C. \end{align*} Hence \[ \eta_{13} \leq C \eta_{10}\leq C \|h\|_{MK^{\alpha_{+},\lambda_{1}}_{q_{1}(\cdot),p(\cdot)}(\mathbb{R}^{n})}. \]

This completes the proof Theorem

Theorem 2. Suppose \(b\in BMO(\mathbb{R}^{n})\). Further suppose \(p(\cdot)\in\mathcal{B}(\mathbb{R}^{n}),\;\;q_{1}(\cdot),q_{2}(\cdot) \in\mathcal{P}(\mathbb{R}^{n})\) with \((q_{2})_{-}\geq(q_{1})_{+}\). If \(\lambda_{1}(q_{2})_{+}=\lambda_{2}(q_{1})_{-},\;\;\lambda_{1}/(q_{1})_{-}-n\delta_{12}< \alpha_{+}< \lambda_{1}/(q_{1})_{-}+n\delta_{11}\) with \(\delta_{11},\delta_{12}\) as in Lemma 4, then the commutator \([b,T]\) is bounded from \( MK^{\alpha_{+},\lambda_{1}}_{q_{1}(\cdot),p(\cdot)}(\mathbb{R}^{n})\) to \( MK^{\alpha(\cdot),\lambda_{2}}_{q_{2}(\cdot),p(\cdot)}(\mathbb{R}^{n})\).

Proof. Let \(b\in BMO (\mathbb{R}^{n}),\) and \(h\in MK^{\alpha_{+},\lambda_{1}}_{q_{1}(\cdot),p(\cdot)}(\mathbb{R}^{n})\). We write \[h(x)=\sum\limits_{j=0}^{\infty}h(x)\chi_{j}(x)\triangleq\sum\limits_{j=0}^{\infty}h_{j}(x).\] By the Definition 2, we have \begin{align*} &\|[b,T](h)\|_{MK^{\alpha(\cdot),\lambda_{2}}_{q_{2}(\cdot),p(\cdot)}(\mathbb{R}^{n})}=\inf\left\{ \eta> 0 : \sup\limits_{k_{\circ}\in z } 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}|[b,T](h)\chi_{k}|}{\eta}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p(\cdot)}{q_{2}(\cdot)}}}\leq 1\right\}. \end{align*} Let \begin{align*} \eta_{21}&=\left\|\sum\limits_{j=0}^{k-2}[b,T](h_{j})\right\|_{MK^{\alpha(\cdot),\lambda_{2}}_{q_{2}(\cdot),p(\cdot)}(\mathbb{R}^{n})}=\inf\left\{ \eta> 0 : \sup\limits_{k_{\circ}\in z } 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}\left|\sum\limits_{j=0}^{k-2}[b,T](h_{j})\chi_{k}\right|}{\eta}\right)^{q_{2}(\cdot)} \right\|_{L^{\frac{p(\cdot)}{q_{2}(\cdot)}}}\leq 1\right\},\\ \eta_{22}&=\left\|\sum\limits_{j=k-1}^{k+1}[b,T](h_{j})\right\|_{MK^{\alpha(\cdot),\lambda_{2}}_{q_{2}(\cdot),p(\cdot)}(\mathbb{R}^{n})}=\inf\left\{ \eta> 0 : \sup\limits_{k_{\circ}\in z } 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}\left|\sum\limits_{j=k-1}^{k+1}[b,T](h_{j})\chi_{k}\right|}{\eta}\right)^{q_{2}(\cdot)} \right\|_{L^{\frac{p(\cdot)}{q_{2}(\cdot)}}}\leq 1\right\},\\ \eta_{23}&=\left\|\sum\limits_{j=k+2}^{\infty}[b,T](h_{j})\right\|_{MK^{\alpha(\cdot),\lambda_{2}}_{q_{2}(\cdot),p(\cdot)}(\mathbb{R}^{n})}=\inf\left\{ \eta> 0 : \sup\limits_{k_{\circ}\in z } 2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}\left|\sum\limits_{j=k+2}^{\infty}[b,T](h_{j})\chi_{k}\right|}{\eta}\right)^{q_{2}(\cdot)} \right\|_{L^{\frac{p(\cdot)}{q_{2}(\cdot)}}}\leq 1\right\}. \end{align*} Then, for any \(k_{0}\in \mathbb{Z}\), we deduce that \begin{align*} &2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}}\left\|\left(\frac{2^{k\alpha(\cdot)}|[b,T](h)\chi_{k}|}{\eta}\right)^{q_{2}(\cdot)} \right\|_{L^{\frac{p(\cdot)}{q_{2}(\cdot)}}}\\ &\leq 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}\left|\sum\limits_{j=0}^{\infty}[b,T](h_{j})\chi_{k}\right|}{\sum\limits_{i=1}^{3}\eta_{2i}} \right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p(\cdot)}{q_{2}(\cdot)}}} \leq 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}\left|\sum\limits_{j=0}^{k-2}[b,T](h_{j})\chi_{k}\right|}{\eta_{21}} \right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p(\cdot)}{q_{2}(\cdot)}}} \end{align*} \begin{align*} &+ 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}\left|\sum\limits_{j=k-1}^{k+1}[b,T](h_{j})\chi_{k}\right|}{\eta_{22}} \right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p(\cdot)}{q_{2}(\cdot)}}} + 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}\left|\sum\limits_{j=k+2}^{\infty}[b,T](h_{j})\chi_{k}\right|}{\eta_{23}} \right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p(\cdot)}{q_{2}(\cdot)}}}, \end{align*} and \[\eta=\sum_{i=1}^{3}\eta_{2i}.\] This implies that \[\|[b,T](h)\|_{MK^{\alpha(\cdot),\lambda_{2}}_{q_{2}(\cdot),p(\cdot)}(\mathbb{R}^{n})}\leq C \eta = C \sum_{i=1}^{3}\eta_{2i}.\] Hence, we only need to estimate \[\eta_{21},\eta_{22}\text{and}\eta_{23} \leq C \|b\|_{\ast} \|h\|_{MK^{\alpha(\cdot),\lambda_{1}}_{q_{1}(\cdot),p(\cdot)}(\mathbb{R}^{n})}.\]

Denote \(\eta_{10}\leq C \|h\|_{MK^{\alpha(\cdot),\lambda_{1}}_{q_{1}(\cdot),p(\cdot)}(\mathbb{R}^{n})}\).

Step 1. We estimate \(\eta_{22}\). By the boundedness of commutator \([b,T]\) on \(L^{p(\cdot)}(\mathbb{R}^{n})\), together with Lemma 6, it follows \begin{align*} &2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}\left|\sum\limits_{j=k-1}^{k+1}[b,T](h_{j})\chi_{k}\right|}{\eta_{10}\|b\|_{\ast}}\right)^{q_{2}(\cdot)} \right\|_{L^{\frac{p(\cdot)}{q_{2}(\cdot)}}}\leq 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{\circ}} \left\| \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k-1}^{k+1}[b,T](h_{j})\chi_{k}|}{\eta_{10}\|b\|_{\ast}}\right\|^{(q_{2}^{1})_{k}}_{L^{p(\cdot)}}\\ &\leq 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{\circ}}\left(\sum\limits_{j=k-1}^{k+1} \left\| \frac{2^{(k-j)\alpha_{+}}2^{j\alpha_{+}}|[b,T](h_{j})\chi_{k}|}{\eta_{10}\|b\|_{\ast}}\right\|_{L^{p(\cdot)}}\right)^{(q_{2}^{1})_{k}}\leq 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{\circ}}\left(\sum\limits_{j=k-1}^{k+1} \left\| \frac{2^{j\alpha_{+}}|h_{j}|}{\eta_{10}}\right\|_{L^{p(\cdot)}}\right)^{(q_{2}^{1})_{k}}, \end{align*} where \[ {(q^{1}_{2})k}= \left\{\begin{array}{ll} (q_{2})_{-}, \left\| \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k-1}^{k+1} [b,T](h_{j}) \chi_{k}|}{\eta_{10}\|b\|_{\ast}}\right\|_{L^{{p(\cdot)}}} \leq 1, \\ (q_{2})_{+}, \left\| \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k-1}^{k+1} [b,T](h_{j}) \chi_{k}|}{\eta_{10}\|b\|_{\ast}}\right\|_{L^{{p(\cdot)}}} >1. \end{array}\right. \] Therefore, since \(h\in MK^{\alpha_{+},\lambda_{1}}_{q_{1}(\cdot),p(\cdot)}(\mathbb{R}^{n})\), we can obtain \[2^{-k_{0}\lambda_{1}}\left\|\left( \frac{2^{k\alpha_{+}}|h\chi_{k}|}{\eta_{10}}\right)^{q_{1}(\cdot)}\right\|_{L^{\frac{p(\cdot)}{q_{1}(\cdot)}}}\leq1.\] From this, and by Lemma 6, if \((q_{1})_{+}\leq(q_{2})_{-}\) and \(\lambda_{1}(q_{2})_{+}=\lambda_{2}(q_{1})_{-},\) then we get \begin{align*} &2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k-1}^{k+1}[b,T](h_{j})\chi_{k}|}{\eta_{10}\|b\|_{\ast}}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p(\cdot)}{q_{2}(\cdot)}}}\leq 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}}\left(\sum\limits_{j=k-1}^{k+1} \left\| \frac{2^{j\alpha_{+}}|h_{j}|}{\eta_{10}}\right\|_{L^{p(\cdot)}}\right)^{(q_{2}^{1})_{k}}\\ &\leq 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}}\left( \left\| \frac{2^{k\alpha_{+}}|h\chi_{k}|}{\eta_{10}}\right\|_{L^{p(\cdot)}}\right)^{(q_{2}^{1})_{k}}\leq 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}}\left\|\left( \frac{2^{k\alpha_{+}}|h\chi_{k}|}{\eta_{10}}\right)^{q_{1}(\cdot)}\right\|_{L^{\frac{p(\cdot)}{q_{1}(\cdot)}}}^{\frac{(q_{2}^{1})_{k}}{(q_{1}^{1})_{k}}}\\ &\leq \sum\limits_{k=0}^{k_{0}}\left\{2^{-k_{0}\lambda_{1}}\left\|\left( \frac{2^{k\alpha_{+}}|h\chi_{k}|}{\eta_{10}}\right)^{q_{1}(\cdot)}\right\|_{L^{\frac{p(\cdot)}{q_{1}(\cdot)}}}\right\}^{\frac{(q_{2}^{1})_{k}}{(q_{1}^{1})_{k}}} \end{align*} where \[ {(q^{1}_{1})_{k}}= \left\{\begin{array}{ll} (q_{1})_{-}, \left\| \frac{2^{k\alpha_{+}}|h\chi_{k}|}{\eta_{20}}\right\|_{L^{{p(\cdot)}}} \leq 1, \\ (q_{1})_{+}, \left\|\frac{2^{k\alpha_{+}}|h\chi_{k}|}{\eta_{20}}\right\|_{L^{{p(\cdot)}}} >1. \end{array}\right.\] This implies \begin{equation*}\label{3.25}\eta_{21} \leq C \|b\|_{\ast} \eta_{10} \leq C\|b\|_{\ast} \|h\|_{MK^{\alpha_{+},\lambda_{1}}_{q_{1}(\cdot),p(\cdot)}(\mathbb{R}^{n})}. \end{equation*} Step 2. Next we estimate \(\eta_{22}\). Let \(x\in R_{k},y\in R_{j}\) and \(j \leq k-2\) then \(2|y|< |x|\) and applying the generalized Hölder's inequality, we have \begin{align*} |[b,T]h_{j}(x)|&\leq \int_{R_{j}}|K(x,y)||b(x)-b(y)||h_{j}(y)|dy\leq C 2^{-nk}\int_{R_{j}}|b(x)-b(y)||h_{j}(y)|dy\\ &\leq C 2^{-nk}\left[|b(x)-b_{B_{j}}|\int_{R_{j}}|h_{j}(y)|dy+\int_{R_{j}}|b(y)-b_{B_{j}}||h_{j}(y)|dy\right]\\ &\leq C 2^{-nk}\left[|b(x)-b_{B_{j}}|\|h\|_{L^{1}(\mathbb{R}^{n})}+\|(b-b_{B_{j}})h_{j}\|_{L^{1}(\mathbb{R}^{n})}\right]\,. \end{align*} Therefore, by Lemma 6, we have \begin{align*} &2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=0}^{k-2}[b,T](h_{j})\chi_{k}|}{\|b\|_{\ast}\eta_{10}}\right)^{q_{2}(\cdot)} \right\|_{L^{\frac{p(\cdot)}{q_{2}(\cdot)}}}\leq C 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}} \left\| \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=0}^{k-2}[b,T](h_{j})\chi_{k}|}{\|b\|_{\ast}\eta_{10}}\right\|^{(q^{2}_{2})_{k}}_{L^{p(\cdot)}}\\ &\leq C 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}} \left\| \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=0}^{k-2}2^{(j-k)\varepsilon}2^{-nk}|b(x)-b_{B_{j}}|\|h\|_{L^{1}(\mathbb{R}^{n})}\chi_{k}|}{ \|b\|_{\ast}\eta_{10}}\right\|^{(q^{2}_{2})_{k}}_{L^{p(\cdot)}}\\ &+ C 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}} \left\| \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=0}^{k-2}2^{-nk}\|(b-b_{B_{j}})h_{j}\|_{L^{1}(\mathbb{R}^{n})}\chi_{k}|}{\|b\|_{\ast}\eta_{10}} \right\|^{(q^{2}_{2})_{k}}_{L^{p(\cdot)}}\\ &\leq C 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}}\left(2^{k\alpha_{+}}\sum\limits_{j=0}^{k-2}2^{-nk} \left\|\frac{|(b-b_{j})h_{j}|}{\|b\|_{\ast}\eta_{10}}\right\|_{L^{1}(\mathbb{R}^{n})}\|\chi_{B_{k}}\|_{L^{p(\cdot)}}\right)^{(q^{2}_{2})_{k}}\\ &+ C 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}}\left(2^{k\alpha_{+}}\sum\limits_{j=0}^{k-2}2^{-nk} \left\|\frac{|h_{j}|}{\eta_{10}}\right\|_{L^{1}(\mathbb{R}^{n})}\|b\|_{\ast}^{-1}\|(b-b_{j})\chi_{B_{k}}\|_{L^{p(\cdot)}} \right)^{(q^{2}_{2})_{k}}, \end{align*} where \[{(q^{2}_{2})_{k}}= \left\{\begin{array}{ll} (q_{2})_{-}, \left\| \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=0}^{k-2}[b,T](h_{j}) \chi_{k}|}{\eta_{10}\|b\|_{\ast}}\right\|_{L^{{p(\cdot)}}} \leq 1, \\ (q_{2})_{+}, \left\| \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=0}^{k-2}[b,T](h_{j}) \chi_{k}|}{\eta_{10}\|b\|_{\ast}}\right\|_{L^{{p(\cdot)}}} >1. \end{array}\right.\] By applying Lemmas 3 and 6, we get that \begin{align*} &2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=0}^{k-2}[b,T](h_{j})\chi_{k}|}{\|b\|_{\ast}\eta_{10}}\right)^{q_{2}(\cdot)} \right\|_{L^{\frac{p(\cdot)}{q_{2}(\cdot)}}}\\ &\leq C2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}}\left(2^{k\alpha_{+}}\sum\limits_{j=0}^{k-2} 2^{-nk} \left\|\frac{|h_{j}|}{\eta_{10}}\right\|_{L^{p(\cdot)}(\mathbb{R}^{n})} \left\|\frac{|(b-b_{j})\chi_{B_{j}}|}{\|b\|_{\ast}}\right\|_{L^{p^{\prime}(\cdot)}(\mathbb{R}^{n})}\|\chi_{B_{k}}\|_{L^{p(\cdot)}} \right)^{(q^{2}_{2})_{k}}\\ &+ C 2^{-k_{\circ}\lambda_{2}} \sum\limits_{k=0}^{k_{\circ}}\left(2^{k\alpha_{+}}\sum\limits_{j=0}^{k-2}2^{-nk} \left\|\frac{|h_{j}|}{\eta_{10}}\right\|_{L^{p(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{j}}\|_{L^{p^{\prime}(\cdot)}(\mathbb{R}^{n})}(k-j)\|\chi_{B_{k}}\|_{L^{p(\cdot)}}\right)^{(q^{2}_{2})_{k}}\\ &\leq C 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}}\left(2^{k\alpha_{+}}\sum\limits_{j=0}^{k-2}2^{-nk} \left\|\frac{|h_{j}|}{\eta_{10}}\right\|_{L^{p(\cdot)}(\mathbb{R}^{n})} (k-j)\frac{\|\chi_{B_{j}}\|_{L^{p^{\prime}(\cdot)}}}{\|\chi_{k}\|_{L^{p^{\prime}(\cdot)}}}|B_{k}|\right)^{(q^{2}_{2})_{k}}\\ &\leq C 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}}\left(\sum\limits_{j=0}^{k-2}(k-j)2^{(k-j)(\alpha_{+}-n\delta_{11})} \left\|\frac{2^{j\alpha_{+}}|h_{j}|}{\eta_{10}}\right\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{(q^{2}_{2})_{k}}, \end{align*} Thus, noting that \(h\in MK^{\alpha_{+},\lambda_{1}}_{q_{1}(\cdot),p(\cdot)}(\mathbb{R}^{n}),\lambda_{1}(q_{1})_{-}=\lambda_{2}(q_{2})_{-}\) and \(\alpha_{+}< n\delta_{11}+\lambda_{1}/(q_{1})_{+}\), we obtain \begin{align*} &2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}} \left\| \left(\frac{2^{k\alpha(\cdot)}|\sum\limits_{j=0}^{k-2}[b,T](h_{j})\chi_{k}|}{\|b\|_{\ast}\eta_{10}}\right)^{q_{2}(\cdot)} \right\|_{L^{\frac{p(\cdot)}{q_{2}(\cdot)}}}\\ &\leq C 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}}\left\{\sum\limits_{j=0}^{k-2}(k-j)2^{(k-j)(\alpha_{+}-n\delta_{11})} \left\|\left(\frac{2^{j\alpha_{+}}|h_{j}|}{\eta_{10}}\right)^{q_{1}(\cdot)} \right\|^{\frac{1}{(q^{2}_{1})_{j}}}_{L^{\frac{p(\cdot)}{q_{1}(\cdot)}}(\mathbb{R}^{n})} \right\}^{(q^{2}_{2})_{k}}\\ &\leq C 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}}\left\{\sum\limits_{j=0}^{k-2}(k-j)2^{(k-j)(\alpha_{+}-n\delta_{11})} \left(2^{j\lambda_{1}}2^{-j\lambda_{1}}\sum\limits_{\ell=0}^{j}\left\|\left(\frac{2^{\ell\alpha_{+}}|h\chi_{\ell}|}{\eta_{10}} \right)^{q_{1}(\cdot)}\right\|_{L^{\frac{p(\cdot)}{q_{1}(\cdot)}}(\mathbb{R}^{n})} \right)^{\frac{1}{(q^{2}_{1})_{j}}}\right\}^{(q^{2}_{2})}\\ &\leq C 2^{(k-k_{0})\lambda_{2}} \sum\limits_{k=0}^{k_{0}}\left\{\sum\limits_{j=0}^{k-2}(k-j)2^{(k-j)(\alpha_{+}-n\delta_{11}-\lambda_{1}(q_{1})_{-})} \left(2^{-j\lambda_{1}}\sum\limits_{\ell=0}^{j}\left\|\left(\frac{2^{\ell\alpha_{+}}|h\chi_{\ell}|}{\eta_{10}}\right)^{q_{1}(\cdot)} \right\|_{L^{\frac{p(\cdot)}{q_{1}(\cdot)}}(\mathbb{R}^{n})} \right)^{\frac{1}{(q^{2}_{1})_{j}}}\right\}^{(q^{2}_{2})}\\ &\leq C \sum\limits_{k=0}^{k_{0}}2^{(k-k_{0})\lambda_{2}}\left( \sum\limits_{j=0}^{k-2}(k-j) 2^{(k-j)(\alpha_{+}-n\delta_{11}-\lambda_{1}/(q_{1})_{-})}\right)^{(q^{2}_{2})_{k}}\leq C. \end{align*} where \[{(q^{2}_{1})_{j}}= \left\{\begin{array}{ll} (q_{1})_{-}, \left\| \frac{2^{j\alpha_{+}}|h\chi_{j}|}{\eta_{10}}\right\|_{L^{{p(\cdot)}}} \leq 1, \\ (q_{1})_{+}, \left\|\frac{2^{j\alpha_{+}}|h\chi_{j}|}{\eta_{10}}\right\|_{L^{{p(\cdot)}}} >1. \end{array}\right.\] This implies that \begin{equation*}\label{3.29}\eta_{22} \leq C \|b\|_{\ast} \eta_{10} \leq C\|b\|_{\ast} \|h\|_{MK^{\alpha_{+},\lambda_{1}}_{q_{1}(\cdot),p(\cdot)}(\mathbb{R}^{n})}.\end{equation*} Step 3. Finally, we \(\eta_{23}\). Let \(x\in R_{k},y\in R_{j}\) and \(j \geq k+2\). Since \(\alpha_{+}>-n\delta_{12}+\lambda_{1}/(q_{1})_{-}\), by the similar argument in Step 2, we get \begin{align*} |[b,T]h_{j}(x)|&\leq \int_{R_{j}}|K(x,y)||b(x)-b(y)||h_{j}(y)|dy\leq C 2^{-jn}\left[|b(x)-b_{B_{j}}|\int_{R_{j}}|h_{j}(y)|dy+\int_{R_{j}}|b(y)-b_{B_{j}}||h_{j}(y)|dy\right]\\ &\leq C 2^{-jn}\left[|b(x)-b_{B_{j}}|\|h\|_{L^{1}(\mathbb{R}^{n})}+\|(b-b_{B_{j}})h_{j}\|_{L^{1}(\mathbb{R}^{n})}\right], \end{align*} and \begin{align*} &2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k+2}^{\infty}[b,T](h_{j})\chi_{k}|}{\|b\|_{\ast}\eta_{10}}\right)^{q_{2}(\cdot)} \right\|_{L^{\frac{p(\cdot)}{q_{2}(\cdot)}}}\leq C 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}} \left\| \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k+2}^{\infty}[b,T](h_{j})\chi_{k}|}{\|b\|_{\ast}\eta_{10}}\right\|^{(q^{3}_{2})_{k}}_{L^{p(\cdot)}}\\ &\leq C 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}} \left\| \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k+2}^{\infty}|b(x)-b_{B_{j}}|\|h\|_{L^{1}(\mathbb{R}^{n})}\chi_{k}|}{\|b\|_{\ast}\eta_{10}} \right\|^{(q^{3}_{2})_{k}}_{L^{p(\cdot)}}+ C 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}} \left\| \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k+2}^{\infty}\|(b-b_{B_{j}})h_{j}\|_{L^{1}(\mathbb{R}^{n})}\chi_{k}|}{\|b\|_{\ast}\eta_{10}} \right\|^{(q^{3}_{2})_{k}}_{L^{p(\cdot)}}\\ &\leq C 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}}\left(2^{k\alpha_{+}}\sum\limits_{j=0}^{k-2}2^{-jn} \left\|\frac{|h_{j}|}{\eta_{10}}\right\|_{L^{p(\cdot)}(\mathbb{R}^{n})} (j-k)\frac{\|\chi_{k}\|_{L^{p(\cdot)}}}{\|\chi_{B_{j}}\|_{L^{p(\cdot)}}}|B_{j}|\right)^{(q^{3}_{2})_{k}}, \end{align*} where \[{(q^{3}_{2})_{k}}= \left\{\begin{array}{ll} (q_{2})_{-}, \left\| \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k+2}^{\infty}[b,T](h_{j}) \chi_{k}|}{\eta_{10}\|b\|_{\ast}}\right\|_{L^{{p(\cdot)}}} \leq 1, \\ (q_{2})_{+}, \left\| \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k+2}^{\infty}[b,T](h_{j}) \chi_{k}|}{\eta_{10}\|b\|_{\ast}}\right\|_{L^{{p(\cdot)}}} >1. \end{array}\right.\] Therefore \begin{align*} &2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k+2}^{\infty}[b,T](h_{j})\chi_{k}|}{\|b\|_{\ast}\eta_{10}}\right)^{q_{2}(\cdot)} \right\|_{L^{\frac{p(\cdot)}{q_{2}(\cdot)}}}\\ &\leq C \sum\limits_{k=0}^{k_{0}}2^{(k-k_{0})\lambda_{2}}\left\{\sum\limits_{j=k+2}^{\infty}(j-k) 2^{(k-j)(\alpha_{+}+n\delta_{12}-\lambda_{1}/(q_{1})_{-})}\left(2^{j\lambda_{1}}2^{-j\lambda_{1}}\sum\limits_{\ell=0}^{j}\left\| \left(\frac{2^{\ell\alpha_{+}}|h\chi_{\ell}|}{\eta_{10}}\right)^{q_{1}(\cdot)} \right\|_{L^{\frac{p(\cdot)}{q_{1}(\cdot)}}(\mathbb{R}^{n})} \right)^{\frac{1}{(q^{3}_{1})_{j}}} \right\}^{(q^{3}_{2})_{k}}\\ &\leq C \sum\limits_{k=0}^{k_{0}}2^{(k-k_{0})\lambda_{2}}\left( \sum\limits_{j=k+2}^{\infty}(j-k) 2^{(k-j)(\alpha_{+}+n\delta_{12}-\lambda_{1}/(q_{1})_{-})}\right)^{(q^{3}_{2})_{k}}\\ &\leq C, \end{align*} which implies that \[\eta_{23} \leq C \|b\|_{\ast} \eta_{10} \leq C\|b\|_{\ast} \|h\|_{MK^{\alpha_{+},\lambda_{1}}_{q_{1}(\cdot),p(\cdot)}(\mathbb{R}^{n})}.\] Combining the above estimates for \(\eta_{21},\eta_{22}\) and \(\eta_{23}\), the get our desired result.

Acknowledgments

The author would like to thank referee for her/his carefully reading and helpful comments which led the paper more readable.

Author Contributions

All authors contributed equally in writing of this paper. All authors read and approved the final manuscript.

Conflict of Interests

The authors declare no conflict of interest.