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

Author(s): Omer Abdalrhman1, Afif Abdalmonem2, Shuangping Tao3
1College of Education, Shendi University, Shendi, River Nile State, Sudan.
2Faculty of Science, University of Dalanj, Dalanj, South kordofan, Sudan.
3College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu, P.R. China.
Copyright © Omer Abdalrhman, Afif Abdalmonem, Shuangping Tao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, the boundedness of Calderón-Zygmund operators is obtained on Morrey-Herz spaces with variable exponents \(MK_{q(\cdot),p(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})\) and several norm inequalities for the commutator generated by Calderó-Zygmund operators, BMO function and Lipschitz function are given.

Keywords: Calderón-Zygmund operators, Morrey-Herz spaces, commutators, variable exponent, BMO spaces, Lipschitz spaces.

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
\begin{equation} [b,T]h(x)=b(x)Th(x)-T(bh)(x). \end{equation}
(1)
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 \(0< \delta_{u1},\delta_{u2} 0\) 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

\begin{equation} \label{eq3.5}\|T(h)\|_{MK^{\alpha(\cdot),\lambda_{2}}_{q_{2}(\cdot),p(\cdot)}(\mathbb{R}^{n})}\leq C \eta = C \sum_{i=1}^{3}\eta_{1i}. \end{equation}
(2)
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
\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}T(h_{j})\chi_{k}\right|}{\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-1}^{k+1}T(h_{j})\chi_{k}|}{\eta_{10}}\right\|^{(q_{2}^{1})_{k}}_{L^{p(\cdot)}}\nonumber\\ &\leq 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}}\left(\sum\limits_{j=k-1}^{k+1} \left\| \frac{2^{(k-j)\alpha_{+}}2^{j\alpha_{+}}|T(h_{j})\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(\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}
(3)
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
\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)}}}\nonumber\\ & \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}\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^{2}_{2})_{k}}\nonumber\\ & \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}\times\left\|\frac{h_{j}}{\eta_{10}}\right\|_{L^{p(\cdot)}(\mathbb{R}^{n})} \frac{\|\chi_{B_{j}}\|_{L^{p^{\prime}(\cdot)}}}{\|\chi_{k}\|_{L^{p^{\prime}(\cdot)}}}|B_{k}|\right\}^{(q^{2}_{2})_{k}}\nonumber\\ & \leq C 2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}}\left\{2^{k\alpha_{+}} \sum\limits_{j=0}^{k-2}2^{(j-k)n\delta_{11}}2^{-j\alpha_{+}}\left\|\frac{2^{j\alpha_{+}}h\chi_{j}}{\eta_{10}} \right\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right\}^{(q^{2}_{2})_{k}}\nonumber\\ & \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\|\frac{2^{j\alpha_{+}}h\chi_{j}}{\eta_{10}} \right\|_{L^{p(\cdot)}(\mathbb{R}^{n})} \right\}^{(q^{2}_{2})_{k}}, \end{align}
(4)
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.

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