Boundedness of commutators on herz-morry-hardy spaces with variable exponent

1. Introduction

Suppose \( \mathbb{S}^{n-1}, (n\geq 2)\) denote the unit sphere in \( \mathbb{R}^{n}\) equipped with the normalized Lebesgue measure \( \mathrm{d}\sigma=\mathrm{d}(\sigma^{\prime})\) . Let \( \Omega\) be homogeneous function of degree zero and satisfies The Calderón-Zygmund singular integral operator \( T_{\Omega}\) is defined as Now, we recall the definitions of the corresponding commutators of the Calderón-Zygmund singular integral operator. Suppose that \( b\in \mathrm{BMO}(\mathbb{R}^{n}),\) the commutators \( [b,T_{\Omega}]\) generated by \( b \text{and} T\) is defined as These operators were firstly introduced by Calderón and Zygmund in [1, 2], in which they proved that these operator are bounded on \( L^{p}\) , where \( 0< p< 1\) . Coifmanet al. [3] showed that if \( \Omega\in\dot{\Lambda}_{\gamma}(\mathbb{S}^{n-1})\) where \( \gamma\in (0,1)\) and \( b\in\mathrm{BMO}(\mathbb{R}^{n})\) , then \( [b,T_{\Omega}]\) is bounded on \( L^{p}\) . In 2011, Lu Ding and Yan [4] proved that \( T_{\Omega}\) and the commutator \( [b,T_{\Omega}]\) are bounded on weighted \( (L^{p}(\mathbb{R}^{n}),L^{q}(\mathbb{R}^{n}))\) . In last 30 years, the function spaces with variable exponent have attracted researchers since the paper [5] appeared in 1991, see, for example [6, 7, 8, 9,10] and their references. Recently, Jingshi Xu and Xiaodi Yang [11] studied the Herz-Morrey-Hardy spaces with variable exponent and their applications. Motivated by [11, 12, 13], our main purpose of this paper is to study some boundedness for commutators of Calderón\(-\)Zygmund operators on Herz-Morrey-Hardy space with two variable exponents. The main tools are properties of variable exponent, BMO function and Lipschitz function.

Definition 1. Let \( \Omega\subset\mathbb{R}^{n}\) be a subset of \( \mathbb{R}^{n}\) with the Lebesgue measure \( >0.\) For a measurable function \( p(\cdot):\Omega\rightarrow[1,\infty)\) , the variable Lebesgue space is defined as $$ L^{p(\cdot)}(\Omega):=\left\{h \text{is measurable on} \Omega:\rho_{p}(h) < \infty\right\},$$ where $$ \rho_{p}(h):=\int_{\Omega}\left(\frac{|h(x)|}{\mu}\right)^{p(x)}dx < \infty \text{for some constat} \mu>0.$$ The set \( L^{p(\cdot)}(\Omega)\) is a quasi Banach space with following Luxemburg-Nakano norm $$ \|h\|_{L^{p(\cdot)}}:=\inf\left\{\mu>0 : \rho_{p}(\mu^{-1} h)\leq1\right\}.$$ The space \( L_{\mathrm{loc}}^{p(\cdot)}(\Omega)\) is defined as $$ L_{\mathrm{loc}}^{p(\cdot)}(\Omega):=\left\{h: h\chi_{k}\in L^{p(\cdot)}(\mathbb{R}^{n}) \text{for any compact subset} K\subset\Omega\right\}.$$ Suppose \( \mathcal{P}(\Omega)\) represents the set of all function \( p:\Omega\rightarrow[1,\infty)\) . Assume that \( p_{-}=\text{ess}\inf_{x\in\Omega} p(x)\) and \( p_{+}=\text{ess}\sup_{x\in\Omega} p(x)\) . Set \( p_{-}>1 \text{,} p_{+}< \infty\) and \( p(\cdot), p'(\cdot)\) are conjugate exponent function defined by \( 1/p(\cdot)+1/p'(\cdot)=1.\) Let \( \mathcal{B}(\Omega)\) be the set of \( p(\cdot)\in\mathcal{P}(\Omega)\) satisfying that the maximal function is bounded on \( L^{p(\cdot)}\) .

Definition 2. (see[11]. Let \( p(\cdot)\in \mathcal{P}(\mathbb{R}^{n}), 0< q< \infty, 0\leq \lambda < \infty, \alpha(\cdot)\in L^{\infty}(\mathbb{R}^{n})\) and \( N>n+1\) . The homogeneous Herz-Morrey-Hardy spaces \( HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}(\mathbb{R}^{n})\) and nonhomogeneous Herz-Morrey-Hardy spaces \( HMK^{\alpha(\cdot),q}_{p(\cdot),\lambda}(\mathbb{R}^{n})\) are defined as $$ HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}(\mathbb{R}^{n})=\left\{h\in\mathcal{S^{\prime}}(\mathbb{R}^{n}): \|h\|_{HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}(\mathbb{R}^{n})} :=\|G_{N}h\|_{M\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}(\mathbb{R}^{n})}< \infty\right\},$$ $$ HMK^{\alpha(\cdot),q}_{p(\cdot),\lambda}(\mathbb{R}^{n})=\left\{h\in\mathcal{S^{\prime}}(\mathbb{R}^{n}):\|h\|_{HMK^{\alpha(\cdot),q}_{p(\cdot),\lambda}(\mathbb{R}^{n})} :=\|G_{N}h\|_{M\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}(\mathbb{R}^{n})}< \infty\right\}.$$ respectively.

2. Preliminaries and Lemmas

Proposition 3. (see[14]. Given a function \( p(\cdot): \mathbb{R}^{n} \rightarrow [ 1 , \infty).\) If \( p(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\) satisfies

and then, \( p(\cdot)\in \mathfrak{B}(\mathbb{R}^{n})\) .

Lemma 4. (see[5]). (Generalized Hölder's Inequality) Given \( p(\cdot), p_{1}(\cdot), p_{2}(\cdot)\in\mathcal{P}(\mathbb{R}^{n})\) .

1. For every \( h\in L^{p_{1}(\cdot)}(\mathbb{R}^{n}) \text{and} g\in L^{p_{2}(\cdot)}(\mathbb{R}^{n})\) , we have $$ \int_{\mathbb{R}^{n}}|h(x) g(x)| dx \leq C\|h\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\|g\|_{L^{p'(\cdot)}(\mathbb{R}^{n})},$$ where \( C_{p}=1+\frac{1}{p_{-}}-\frac{1}{p_{+}}\) .
2. For every \( h\in L^{p_{1}(\cdot)}(\mathbb{R}^{n}),\) and \( g\in L^{p_{2}(\cdot)}(\mathbb{R}^{n})\) , when \( \frac{1}{p(\cdot)}=\frac{1}{p_{2}(\cdot)}+\frac{1}{p_{1}(\cdot)}\) , we have $$ \|h(x)g(x)\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\leq C \|g(x)\|_{L^{p_{2}}(\mathbb{R}^{n})}\|h(x)\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})},$$ where \( C_{p_{1},p_{2}}=[1+\frac{1}{p_{1-}}-\frac{1}{p_{1+}}]^{\frac{1}{p_{-}}}\) .

Lemma 5. (see[15, 16]). Given \( p(\cdot) \in \mathcal{B}(\mathbb{R}^{n})\) . If there exist positive constants \( C,\) \( \delta_{1}\) and \( \delta_{2}\) such that \( \delta_{1}, \delta_{2}< 1\) , then for all balls \( B\subset\mathbb{R}^{n}\) and all measurable subset \( R\subset B,\) we have $$ \frac{\|\chi_{R}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}}{\|\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}}\leq C \frac{|R|}{|B|}, \frac{\|\chi_{R}\|_{L^{p^{\prime}(\cdot)}(\mathbb{R}^{n})}}{\|\chi_{B}\|_{L^{p^{\prime}(\cdot)}(\mathbb{R}^{n})}}\leq C\left(\frac{|R|}{|B|}\right)^{\delta_{2}}, \frac{\|\chi_{R}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}}{ \|\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}}\leq C\left(\frac{|R|}{|B|}\right)^{\delta_{1}}.$$

Lemma 6. (see[17]). If \( p(\cdot) \in \mathcal{B}(\mathbb{R}^{n}),\) then there exists a constant \( C > 0\) such that for any ball \( B\) in \( \mathbb{R}^{n}\) , we have $$ \frac{1}{|B|}\|\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})} \|\chi_{B}\|_{L^{p'(\cdot)}(\mathbb{R}^{n})}\leq C. $$ Now, the BMO function and BMO norm are defined as \begin{align*} \mathrm{BMO}(\mathbb{R}^{n})&:=\left\{b\in L^{1}_{loc}(\mathbb{R}^{n}):\|b\|_{\mathrm{BMO}(\mathbb{R}^{n})}< 0\right\},\\ \|b\|_{\mathrm{BMO}(\mathbb{R}^{n})}&:=\sup\limits_{Q:\text{cube}}|Q|^{-1}\int_{Q}|b(x)-b_{Q}|\text{d}x. \end{align*} respectively.

Lemma 7. (see[18]). Given \( p(\cdot) \in \mathcal{B}(\mathbb{R}^{n}), b\in \mathrm{BMO}(\mathbb{R}^{n})\) . If \( i,j\in\mathbb{Z}\) with \( i< j\) , then we have

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

Lemma 8. (see[18]). Suppose that \( p(\cdot)\in\mathcal{P}(\mathbb{R}^{n}), q\in[0,\infty)\) and \( \lambda\in[0,\infty)\) . If \( \alpha(\cdot)\) is log-Hölder's continuous both at origin and at infinity, then \begin{eqnarray*} &&\qquad\|h\|_{M\dot{K}_{p(\cdot),\lambda}^{\alpha(\cdot),q}(\mathbb{R}^{n})}\approx max \left\{{\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda } \left(\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q} \left\|h\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{1/q}}, \right.\\ &&\left. \sup\limits_{L>0,L\in \mathbb{Z}} \left[2^{-L\lambda } \left(\sum\limits_{k=-\infty}^{-1} 2^{k\alpha(0)q}\left\|h\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{1/q} + 2^{-L\lambda }\left( \sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q} \left\|h\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{1/q}\right]\right\}. \end{eqnarray*}

Lemma 9. (see[4]). Let \( \Omega\) satisfies \( L^{r}\) -Dini condition \( r\in[1,\infty)\) . If there exist constants \( C>0\) and \( R>0\) such that \( |y|< R/2\) , then for every \( x\in\mathbb{R}^{n}\) , we have $$ \left(\int_{R< |x|< 2R}\left|\frac{\Omega(x-y)}{|x-y|^{n}}-\frac{\Omega(x)}{|x|^{n}}\right|^{r}dx\right)^{\frac{1}{r}} \leq CR^{(\frac{n}{r}-n)}\left\{\frac{|y|}{R}+\int_{|y|/2R< \delta< |y|/R}\frac{w_{r}(\delta)}{\delta}d\delta\right\}.$$

Lemma 10. (see[11]). Suppose that \( p(\cdot)\in\mathcal{B}(\mathbb{R}^{n}), q\in[0,\infty)\) and \( \lambda\in[0,\infty)\) . Let \( \alpha(\cdot)\) is log-Hölder's continuous both at origin and at infinity. If \( 2\lambda\leq \alpha(\cdot), n\delta_{2}\leq \alpha(0), \alpha< \infty \text{and} \delta_{2}\) as defined in Lemma \ref{l2.2}. Then \( h\in HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}(\mathbb{R}^{n}) \left(\text{or} HM{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}(\mathbb{R}^{n})\right)\) if and only if \( h=\sum\limits_{k=-\infty}^{\infty}\lambda_{k}g_{k} \left(\text{or} \sum\limits_{k=0}^{\infty}\lambda_{k}g_{k}\right)\) , in the sense of \( \mathcal{S}'(\mathbb{R}^{n})\) , where each \( g_{k}\) be a central \( (\alpha(\cdot),p(\cdot))\) -atom (or central \( (\alpha(\cdot), p(\cdot))\) -atom of restricted type) with support contained in \( B_{k}\) and \( \sup\limits_{L\in\mathbb{Z}} 2^{-L\lambda}\sum\limits_{k=-\infty}^{L}|\lambda_{k}|^{q} < \infty\) or \( \left(\sup\limits_{L\in\mathbb{Z}} 2^{-L\lambda}\sum\limits_{k=0}^{L}|\lambda_{k}|^{q}< \infty\right).\) \\ Also,\\ \( \|h\|_{HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}}\approx\inf\sup\limits_{L\in\mathbb{Z}} 2^{L\lambda} \left(\sum\limits_{k=-\infty}^{L}|\lambda_{k}|^{q}\right)^{1/q}\left(\text{or} \|h\|_{HM{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}}\approx\inf\sup\limits_{L\in\mathbb{Z}} 2^{L\lambda} \left(\sum\limits_{k=0}^{L}|\lambda_{k}|^{q}\right)^{1/q}\right),\) \\ where infimum is taken over all above decomposition of \( h\) .

Lemma 11(see [19]). Let \( p(\cdot)\in \mathcal{P}(\Omega)\) and \( h:\Omega\times \Omega\rightarrow \mathbb{R}\) is a measurable function (with respect to product measure) such that, for almost every \( y\in \Omega, h(\cdot,y)\in L^{p(\cdot)}(\Omega)\) . Then $$ \left\|\int_{\Omega}h(\cdot,y)dy\right\|_{L^{p(\cdot)}(\Omega)}\leq C \int_{\Omega}\left\|h(\cdot,y)\right\|_{L^{p(\cdot)}(\Omega)}dy.$$

Lemma 12. (see[19]). Suppose \( p(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\) satisfies conditions \eqref{2.1} and \eqref{2.2} of Proposition \ref{p2.1}, then for any ball (or cube) \( Q\subset\mathbb{R}^{n}\) , we have $$ {\|\chi_{Q}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}}\approx\left\{\begin{array}{ll} |Q|^{\frac{1}{p(x)}}, \text{if} |Q|\leq 2^{n};\\ |Q|^{\frac{1}{p(\infty)}}, \text{if} |Q|\geq 1, \end{array}\right.$$ where \( p(\infty)=\lim\limits_{x\rightarrow\infty}p(x).\)

3. Main Results

In this section, we formulate and prove the main results of this paper.

Theorem 13. Let \( p(\cdot)\in \mathcal{B}(\mathbb{R}^{n}) \text{and} \Omega\in L^{r}(\mathbb{S}^{n-1})(r>p^{+})\) satisfies

Suppose that \( 0< q< \infty, 0\leq\lambda< \infty \text{and} \alpha(\cdot)\in L^{\infty}(\mathbb{R}^{n})\) satisfies conditions (4) and (5) of Proposition 3. If \( 2\lambda\leq\alpha(\cdot), n\delta_{2}\leq \alpha(0),\alpha_{\infty}< \beta+n\delta_{2},\) then \( T_{\Omega}\) is bounded from \( HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}\) or \( \left(HMK^{\alpha(\cdot),q}_{p(\cdot),\lambda}\right)\) to \( M\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}\) or \( \left(MK^{\alpha(\cdot),q}_{p(\cdot),\lambda}\right).\)

Proof. It suffices to prove for \( HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}\) . Assume that \( h\in HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}\) , then by Lemma 10, \( h=\sum\limits_{j=-\infty}^{\infty}\lambda_{j}g_{j}\) converges in \( \mathcal{S}'(\mathbb{R}^{n}),\) where \( \|h\|_{HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}}\approx\inf\sup\limits_{L\in\mathbb{Z}} 2^{L\lambda} (\sum\limits_{j=-\infty}^{L}|\lambda_{j}|^{q})^{1/q},\) and \( g_{j}\) is a dyadic central \( (\alpha(\cdot),p(\cdot))\) -atom with support contained in \( B_{j}\) . For simplicity, we take \( \Upsilon=\sup\limits_{L\in\mathbb{Z}} 2^{L\lambda}\sum\limits_{j=-\infty}^{L}|\lambda_{j}|^{q}.\) By virtue of Lemma 8, we have \begin{eqnarray*} &&\|T_{\Omega}(h)\|_{M\dot{K}_{p(\cdot),\lambda}^{\alpha(\cdot),q}(\mathbb{R}^{n})}\approx max \left\{{\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda } \left(\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q} \left\|T_{\Omega}(h)\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{1/q}}, \right.\\ &&\left. \sup\limits_{L>0,L\in \mathbb{Z}} \left[2^{-L\lambda } \left(\sum\limits_{k=-\infty}^{-1} 2^{k\alpha(0)q}\left\|T_{\Omega}(h)\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{1/q} + 2^{-L\lambda }\left( \sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q} \left\|T_{\Omega}(h)\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{1/q}\right]\right\}\\ &&\approx max \left\{E,F+G\right\}. \end{eqnarray*} Let \begin{eqnarray*} E&=&\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q}\left\|T_{\Omega}(h)\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})},\\ F&=&\sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\left\|T_{\Omega}(h)\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})},\\ G&=&\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q} \left\|T_{\Omega}(h)\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})}. \end{eqnarray*} To finish our proof, we only need to show that there exists a constant \( C>0\) , such that \( E, F \text{,} G\leq C\Upsilon.\)
First we prove that \( E\leq C\Upsilon\) . \begin{eqnarray*} E&=&\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q}\left\|T_{\Omega}(h)\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})}\leq \sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q}\left(\sum\limits_{j=k}^{\infty}|\lambda_{j}| \|T_{\Omega}g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\\ && +\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}| \|T_{\Omega}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}:=E_{1}+E_{2}. \end{eqnarray*} By the \( \left(L^{p(\cdot)}(\mathbb{R}^{n}),L^{p(\cdot)}(\mathbb{R}^{n})\right)\) -boundedness of the \( T_{\Omega}\) (see[13]), we get $$ \|T_{\Omega}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\leq \|g_{j}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\leq |B_{j}|^{-\alpha_{j}/n}=2^{-j\alpha_{j}}.$$ Therefore, when \( 0< q\leq1,\) we obtain \begin{eqnarray*} E_{1}&=&\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q}\left(\sum\limits_{j=k}^{\infty}|\lambda_{j}| \|T_{\Omega}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\\ &\leq& C \sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q}\left(\sum\limits_{j=k}^{\infty}|\lambda_{j}| 2^{-j\alpha_{j}}\right)^{q}\\ &\leq& C \sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q}\left[\left(\sum\limits_{j=k}^{-1}|\lambda_{j}| 2^{-j\alpha(0)}+\sum\limits_{j=0}^{\infty}|\lambda_{j}|2^{-j\alpha_{\infty}}\right)^{q}\right]\\ &\leq& C \left[\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}\sum\limits_{j=k}^{-1}|\lambda_{j}|^{q} 2^{(k-j)\alpha(0)q}\right.\left.+\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\left( \sum\limits_{k=-\infty}^{L}2^{k\alpha(0)}\sum\limits_{j=0}^{\infty}|\lambda_{j}|2^{-j\alpha_{\infty}} \right)^{q}\right]\\ &\leq& C \left[\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{j=-\infty}^{-1}|\lambda_{j}|^{q}\sum\limits_{k=-\infty}^{j} 2^{(k-j)\alpha(0)q}\right.\left.+\sup\limits_{L\leq0,L\in \mathbb{Z}} \sum\limits_{j=0}^{\infty}2^{-j\lambda q} |\lambda_{j}|^{q}2^{(\lambda-\alpha_{\infty})jq}2^{-L\lambda q}\sum\limits_{k=\infty}^{L}2^{k\alpha(0)q}\right]\\ &\leq& C \left[\sup\limits_{L\leq0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{j=-\infty}^{L}|\lambda_{j}|^{q} +\sup\limits_{L\leq0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{j=L}^{-1}|\lambda_{j}|^{q}\sum\limits_{k=-\infty}^{j}2^{(k-j)\alpha(0)q}\right.\\ && +\left.\Upsilon\sup\limits_{L\leq0,L\in \mathbb{Z}}\sum\limits_{j=0}^{\infty}2^{(\lambda-\alpha_{\infty})jq}\sum\limits_{k=\infty}^{L}2^{[k\alpha(0)-L\lambda]q}\right]\\ &\leq& C \left[\Upsilon+\sup\limits_{L\leq0,L\in \mathbb{Z}}\sum\limits_{j=L}^{-1}2^{-j\lambda q}|\lambda_{j}|^{q}2^{(j-L)\lambda q}\sum\limits_{k=-\infty}^{j}2^{(k-j)\alpha(0)q}+\Upsilon\right]\\ &\leq& C \left[\Upsilon+\Upsilon\sup\limits_{L\leq0,L\in \mathbb{Z}}\sum\limits_{j=L}^{-1}2^{(j-L)\lambda q}\sum\limits_{k=-\infty}^{j}2^{(k-j)\alpha(0)q}+\Upsilon\right]\leq C \Upsilon. \end{eqnarray*} when \( 1< q< \infty\) , and \( 1/q+1/q^{\prime}=1\) , we have \begin{eqnarray*} E_{1}&=&\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q}\left(\sum\limits_{j=k}^{\infty}|\lambda_{j}| \|T_{\Omega}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\\ &\leq& C \sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q}\left(\sum\limits_{j=k}^{\infty}|\lambda_{j}| 2^{-j\alpha_{j}}\right)^{q}\\ &\leq& C \left[\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}\left(\sum\limits_{j=k}^{-1}|\lambda_{j}| 2^{(k-j)\alpha(0)}\right)^{q}\right.\left.+\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q}\left(\sum\limits_{j=0}^{\infty}|\lambda_{j}|2^{-j\alpha_{\infty}} \right)^{q}\right]\\ &\leq& C \left[\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}\left(\sum\limits_{j=k}^{-1}|\lambda_{j}|^{q} 2^{(k-j)\alpha(0){\frac{q}{2}}}\right)\times\left(\sum\limits_{j=k}^{-1} 2^{(k-j)\alpha(0){\frac{q^{\prime}}{2}}}\right)^{\frac{q}{q^{\prime}}}\right.\\ && \left.+\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q}\left(\sum\limits_{j=0}^{\infty} |\lambda_{j}|^{q}2^{-j\alpha_{\infty}\frac{q}{2}}\right) \times\left(\sum\limits_{j=0}^{\infty}2^{-j\alpha_{\infty}\frac{q^{\prime}}{2}} \right)^{\frac{q}{q^{\prime}}}\right]\\ &\leq& C \left[\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}\sum\limits_{j=k}^{-1}|\lambda_{j}|^{q} 2^{(k-j)\alpha(0){\frac{q}{2}}}\right.\left.+\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q}\sum\limits_{j=0}^{\infty}|\lambda_{j}|^{q} 2^{-j\alpha_{\infty}\frac{q}{2}}\right]\\ &\leq& C \left[\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{j=-\infty}^{-1}|\lambda_{j}|^{q}\sum\limits_{k=\infty}^{j} 2^{(k-j)\alpha(0){\frac{q}{2}}}\right.\left.+\sup\limits_{L\leq0,L\in \mathbb{Z}}\sum\limits_{j=0}^{\infty}2^{-j\lambda q}|\lambda_{j}|^{q} 2^{(\lambda-\frac{\alpha_{\infty}}{2})jq}2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q}\right]\\ &\leq& C \left[\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{j=-\infty}^{L}|\lambda_{j}|^{q} +\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{j=L}^{-1}|\lambda_{j}|^{q}\sum\limits_{k=\infty}^{j} 2^{(k-j)\alpha(0){\frac{q}{2}}}\right.\\ && \left.+\Upsilon\sup\limits_{L\leq0,L\in \mathbb{Z}}\sum\limits_{j=0}^{\infty} 2^{(\lambda-\frac{\alpha_{\infty}}{2})jq}2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q}\right] \end{eqnarray*} \begin{eqnarray}\label{3.1} &&\leq C \left[\Upsilon+\sup\limits_{L\leq0,L\in \mathbb{Z}} \sum\limits_{j=L}^{-1}2^{-j\lambda q}|\lambda_{j}|^{q}2^{(j-L)\lambda q}\sum\limits_{k=\infty}^{j} 2^{(k-j)\alpha(0){\frac{q}{2}}}+\Upsilon\right]\nonumber\\ &&\leq C \left[\Upsilon+\Upsilon\sup\limits_{L\leq0,L\in \mathbb{Z}} \sum\limits_{j=L}^{-1}2^{(j-L)\lambda q}\sum\limits_{k=\infty}^{j} 2^{(k-j)\alpha(0){\frac{q}{2}}}+\Upsilon\right]\nonumber\\ &&\leq C \Upsilon. \end{eqnarray} Now, we prove that \( E_{2}\leq C\Upsilon\) . Note that if \( x\in A_{k} \text{for each} k\in\mathbb{Z},y\in A_{j}\) and \( j\leq k-1\) . Let \( \tilde{p}(\cdot)>1 \text{and} 1/p(\cdot)=1/\tilde{p}(\cdot)+1/r\) . Since \( r>p^{+}\) , so by Lemma 4 and Lemma 11, we get

Using Lemma 9, we get By Lemma 12, we obtain From (8) and Lemmas 4-6, we get So, when \( 0< q\leq1,\) we obtain When \( 0< q< \infty,\) and \( 1/q+1/q^{\prime}=1,\) by \( n\delta_{2}\leq \alpha(0) < \beta +n\delta_{2}\) and Hölder inequality, we have Next we prove that \( F\leq C\Upsilon.\) Since \( 0< q\leq1\) , we get when \( 0< q< \infty\) and \( 1/q+1/q^{\prime}=1,\) we deduce \begin{eqnarray*} F_{1}&=&\sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\left(\sum\limits_{j=k}^{\infty}|\lambda_{j}| \|T_{\Omega}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\leq C \sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\left(\sum\limits_{j=k}^{\infty}|\lambda_{j}| 2^{-j\alpha_{j}}\right)^{q}\\ &\leq& C \left[\sum\limits_{k=-\infty}^{-1}\left(\sum\limits_{j=k}^{-1}|\lambda_{j}| 2^{(k-j)\alpha(0)}\right)^{q} +\sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\left(\sum\limits_{j=0}^{\infty} |\lambda_{j}|2^{-j\alpha_{\infty}}\right)^{q}\right]\\ &\leq& C \left[\sum\limits_{k=-\infty}^{-1}\left(\sum\limits_{j=k}^{-1}|\lambda_{j}|^{q} 2^{(k-j)\alpha(0){\frac{q}{2}}}\right)\times\left(\sum\limits_{j=k}^{-1} 2^{(k-j)\alpha(0){\frac{q^{\prime}}{2}}}\right)^{\frac{q}{q^{\prime}}}\right.\\ && \left.+\sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\left(\sum\limits_{j=0}^{\infty} |\lambda_{j}|^{q}2^{-j\alpha_{\infty}\frac{q}{2}}\right) \times\left(\sum\limits_{j=0}^{\infty}2^{-j\alpha_{\infty}\frac{q^{\prime}}{2}}\right)^{\frac{q}{q^{\prime}}} \right] \end{eqnarray*} \begin{eqnarray*} && \leq C \left[\sum\limits_{k=-\infty}^{-1}\sum\limits_{j=k}^{-1}|\lambda_{j}|^{q} 2^{(k-j)\alpha(0){\frac{q}{2}}} + \sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\sum\limits_{j=0}^{\infty} |\lambda_{j}|^{q}2^{-j\alpha_{\infty}\frac{q}{2}}\right]\\ && \leq C \left[\sum\limits_{j=-\infty}^{-1}|\lambda_{j}|^{q}\sum\limits_{k=\infty}^{j} 2^{(k-j)\alpha(0){\frac{q}{2}}}+\sum\limits_{j=0}^{\infty}2^{-j\lambda q}\sum\limits_{l=-\infty}^{j}|\lambda_{l}|^{q} 2^{(\lambda-\frac{\alpha_{\infty}}{2})jq} \sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\right]\\ && \leq C \left[\Upsilon\sum\limits_{k=\infty}^{j} 2^{(k-j)\alpha(0){\frac{q}{2}}}+\Upsilon\sum\limits_{j=0}^{\infty} 2^{(\lambda-\frac{\alpha_{\infty}}{2})jq} \sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\right]\nonumber\\ &&\leq C \Upsilon. \end{eqnarray*} Now we prove that \( F_{2}\leq C \Upsilon.\) When \( 0< q\leq1\) , from (11) and \( n\delta_{2}\leq \alpha(0) < \beta+n\delta_{2}\) , we have \begin{eqnarray*} F_{2}&=&\sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}| \|T_{\Omega}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\leq C \sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}| 2^{(j-k)(\beta+n\delta_{2})-j\alpha(0)}\right)^{q}\nonumber\\ &&\leq C \sum\limits_{k=-\infty}^{-1}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}| 2^{[(j-k)(\beta+n\delta_{2})-\alpha(0)]}\right)^{q}\leq C \sum\limits_{j=-\infty}^{-1}|\lambda_{j}|^{q}\left(\sum\limits_{k=j+1}^{-1} 2^{[(j-k)(\beta+n\delta_{2})-\alpha(0)]}\right)^{q}\nonumber\\ &&\leq C \Upsilon. \end{eqnarray*} when \( 0< q\leq\infty,\) and \( 1/q+1/q^{\prime}=1,\) by \( n\delta_{2}\leq \alpha(0) < \beta+n\delta_{2}\) and Hölder inequality, we obtain Finally we prove that \( G\leq C\Upsilon.\) \begin{eqnarray*} &G&=\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q} \left\|T_{\Omega}(h)\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})}\leq \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q}\left(\sum\limits_{j=k}^{\infty}|\lambda_{j}| \|T_{\Omega}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\\ && +\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}| \|T_{\Omega}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\\ &&:=G_{1}+G_{2}. \end{eqnarray*} When \( 0< q\leq1,\) by the boundedness of the commutator \( [b,T]\) in \( L^{p(\cdot)}(\mathbb{R}^{n})\) , we obtain When \( 0< q< \infty,\) by using Hölder inequality, we have \begin{eqnarray*} &G_{1}&=\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q}\left(\sum\limits_{j=k}^{\infty}|\lambda_{j}| \|T_{\Omega}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\\ && \leq C \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q}\left(\sum\limits_{j=k}^{\infty}|\lambda_{j}|^{q} |B_{j}|^{-j\alpha_{j}\frac{q}{2n}}\right)\times\left(\sum\limits_{j=k}^{\infty} |B_{j}|^{-j\alpha_{j}\frac{q^{\prime}}{2n}}\right)^{\frac{q}{q^{\prime}}}\\ && \leq C \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}\left(\sum\limits_{j=k}^{\infty}|\lambda_{j}|^{q} 2^{(k-j)\alpha_{\infty}\frac{q}{2}}\right)\\ && \leq C \left[\sup\limits_{L>0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{j=0}^{L}|\lambda_{j}|^{q}\sum\limits_{k=0}^{j}2^{(k-j)\alpha_{\infty}\frac{q}{2}} +\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{j=L}^{\infty}|\lambda_{j}|^{q}\sum\limits_{k=0}^{L}2^{(k-j)\alpha_{\infty}\frac{q}{2}}\right]\\ && \leq C \left[\sup\limits_{L>0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{j=0}^{L}|\lambda_{j}|^{q} +\sup\limits_{L>0,L\in \mathbb{Z}}\sum\limits_{j=L}^{\infty}2^{(j-L)\lambda q}2^{-j\lambda q}\sum\limits_{l=-\infty}^{j}|\lambda_{l}|^{q}\sum\limits_{k=0}^{L}2^{(k-j)\alpha_{\infty}\frac{q}{2}}\right]\\ && \leq C \left[\Upsilon +\Upsilon\sup\limits_{L>0,L\in \mathbb{Z}}\sum\limits_{j=L}^{\infty}2^{(j-L)\lambda q}2^{(L-j)\frac{\alpha_{\infty}}{2}q}\right]\\ && \leq C \left[\Upsilon +\Upsilon\sup\limits_{L>0,L\in \mathbb{Z}}\sum\limits_{j=L}^{\infty}2^{(j-L)(\lambda-\frac{\alpha_{\infty}}{2}) q}\right]\nonumber\\ && \leq C \Upsilon. \end{eqnarray*} For \( G_{2}\leq C \Upsilon\) , when \( 0< q< \infty,\) and \( 1/q+1/q^{\prime}=1,\) from (7), \( n\delta_{2} \leq\alpha(0), \alpha_{\infty}< \beta +n\delta_{2}\) and applying Hölder inequality, we have \begin{eqnarray*} G_{2}&=&\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}| \|T_{\Omega}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\\ &\leq& C \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}| 2^{(j-k)(\beta+n\delta_{2})-j\alpha_{j}}\right)^{q}\\ &\leq& C \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q}\left(\sum\limits_{j=-\infty}^{-1}|\lambda_{j}| 2^{(j-k)(\beta+n\delta_{2})-j\alpha(0)}\right)^{q}\\ && + C \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q}\left(\sum\limits_{j=0}^{k-1}|\lambda_{j}| 2^{(j-k)(\beta+n\delta_{2})-j\alpha_{\infty}}\right)^{q}\\ &\leq& C \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{[\alpha_{\infty}-(\beta+n\delta_{2})]kq}\left(\sum\limits_{j=-\infty}^{-1}|\lambda_{j}| 2^{[(\beta+n\delta_{2})-\alpha(0)]j}\right)^{q}\\ && + C \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}\left(\sum\limits_{j=0}^{k-1}|\lambda_{j}| 2^{(j-k)[(\beta+n\delta_{2})-\alpha_{\infty}]}\right)^{q}\\ &\leq& C \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{[\alpha_{\infty}-(\beta+n\delta_{2})]kq}\left(\sum\limits_{j=-\infty}^{-1}|\lambda_{j}| 2^{j\frac{[(\beta+n\delta_{2})-\alpha(0)]}{2}}\right)^{q} \times\left(\sum\limits_{j=-\infty}^{-1}(k-j)^{q^{\prime}} 2^{j[(\beta+n\delta_{2})-\alpha(0)]\frac{q^{\prime}}{2}}\right)^{\frac{q}{q^{\prime}}}\\ && + C \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}\left(\sum\limits_{j=0}^{k-1}|\lambda_{j}| 2^{(j-k)\frac{[(\beta+n\delta_{2})-\alpha_{\infty}]}{2}}\right)^{q} \times\left(\sum\limits_{j=0}^{k-1}(k-j)^{q^{\prime}} 2^{(j-k)[(\beta+n\delta_{2})-\alpha_{\infty}]\frac{q^{\prime}}{2}}\right)^{\frac{q}{q^{\prime}}}\\ &\leq& C \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{j=-\infty}^{-1}|\lambda_{j}|^{q} 2^{[(\beta+n\delta_{2})-\alpha(0)]j\frac{q}{2}} + C \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}\sum\limits_{j=0}^{k-1}|\lambda_{j}|^{q} 2^{(j-k)[(\beta+n\delta_{2})-\alpha_{\infty}]\frac{q}{2}}\\ &\leq& C \left[\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{j=-\infty}^{-1}|\lambda_{j}|^{q} \right.\left.+ \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}\sum\limits_{j=0}^{k-1}|\lambda_{j}|^{q} 2^{(j-k)[(\beta+n\delta_{2})-\alpha_{\infty}]\frac{q}{2}}\right]\nonumber\\ \end{eqnarray*} \begin{eqnarray*} &\leq& C \left[\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{j=-\infty}^{-1}|\lambda_{j}|^{q} \right.\left.+ \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{j=0}^{L-1}|\lambda_{j}|^{q}\sum\limits_{k=j+1}^{L} 2^{(j-k)[(\beta+n\delta_{2})-\alpha_{\infty}]\frac{q}{2}}\right]\nonumber\\ &\leq& C \left[\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{j=-\infty}^{-1}|\lambda_{j}|^{q} + \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{j=0}^{L-1}|\lambda_{j}|^{q}\right]\nonumber\\ &\leq& C \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{j=-\infty}^{L-1}|\lambda_{j}|^{q}\leq C \Upsilon. \end{eqnarray*} The proof is completed.

Theorem 14. Suppose that \( b\in \mathrm{BMO}(\mathbb{R}^{n}), p(\cdot)\in \mathcal{B}(\mathbb{R}^{n})\) and let \(\Omega\in L^{r}(\mathbb{S}^{n-1})(r>p^{+})\) satisfies (7). Let \( 0< q< \infty, 0\leq\lambda< \infty \text{and} \alpha(\cdot)\in L^{\infty}(\mathbb{R}^{n})\) satisfies conditions (4) and (5) of Proposition 3. If \( 2\lambda\leq\alpha(\cdot), n\delta_{2}\leq \alpha(0),\alpha_{\infty}< \beta+n\delta_{2},\) then \( [b,T_{\Omega}]\) is bounded from \( HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}\) or \( \left(HMK^{\alpha(\cdot),q}_{p(\cdot),\lambda}\right)\) to \( M\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}\) or \( \left(MK^{\alpha(\cdot),q}_{p(\cdot),\lambda}\right).\)

Proof. It suffices to prove for \( HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}\) . Set \( b\in \mathrm{BMO}(\mathbb{R}^{n})\) and \( h\in HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}\) . By Lemma 10, \( h=\sum\limits_{j=-\infty}^{\infty}\lambda_{j}g_{j}\) converges in \( \mathcal{S}'(\mathbb{R}^{n}),\) where \( \|h\|_{HM\dot{K}^{\alpha(\cdot),q}_{p(\cdot),\lambda}}\approx\inf\sup\limits_{L\in\mathbb{Z}} 2^{L\lambda} \left(\sum\limits_{j=-\infty}^{L}|\lambda_{j}|^{q}\right)^{1/q},\) and \( g_{j}\) is a dyadic central \( (\alpha(\cdot),p(\cdot))\) -atom with support contained in \( B_{j}\) . For simplicity, we denote \( \Upsilon=\sup\limits_{L\in\mathbb{Z}} 2^{L\lambda}\sum\limits_{j=-\infty}^{L}|\lambda_{j}|^{q}.\) By virtue of Lemma 8, we rewrite $$ \begin{array}{ll} \|T^{b}_{\Omega}(h)\|_{M\dot{K}_{p(\cdot),\lambda}^{\alpha(\cdot),q}(\mathbb{R}^{n})} \approx max \left\{{\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda } \left(\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q} \left\|T^{b}_{\Omega}(h)\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{1/q}}, \right.\\ \\ \left. \sup\limits_{L>0,L\in \mathbb{Z}} \left[2^{-L\lambda } \left(\sum\limits_{k=-\infty}^{-1} 2^{k\alpha(0)q}\left\|T^{b}_{\Omega}(h)\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{1/q} + 2^{-L\lambda }\left( \sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q} \left\|T^{b}_{\Omega}(h)\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{1/q}\right]\right\}\\ \approx max \left\{E',F'+G'\right\} \end{array}$$ where \begin{eqnarray*} &E'&=\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q}\left\|T^{b}_{\Omega}(h)\chi_{k} \right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})},\\ &F'&=\sum\limits_{k=-\infty}^{-1} 2^{k\alpha(0)q}\left\|T^{b}_{\Omega}(h)\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})},\\ &G'&=\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q} \left\|T^{b}_{\Omega}(h)\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})}. \end{eqnarray*} To complete the prove, we only need to show that there exists a constant \( C>0\) , such that \( E',F' \text{,} G'\leq C\Upsilon.\)
First we show that \( E'\leq C\Upsilon.\) \begin{eqnarray*} &E'&=\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q}\left\|T^{b}_{\Omega}(h)\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})}\\ &&\leq \sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q}\left(\sum\limits_{j=k}^{\infty}|\lambda_{j}| \|T^{b}_{\Omega}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\\ && +\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}| \|T^{b}_{\Omega}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\\ &&:=E'_{1}+E'_{2}. \end{eqnarray*} By the \( \left(L^{p(\cdot)}(\mathbb{R}^{n}),L^{p(\cdot)}(\mathbb{R}^{n})\right)\) -boundedness of the \( T^{b}_{\Omega}\) (see [13]) and following the same way as we estimated \( E_{1}\) in Theorem 13, we get \( E'_{1}\leq C \|b\|_{\ast}\Upsilon.\)
Now, we estimate \( E'_{2}\) . For each \( k\in \mathbb{Z}\) and \( x\in A_{k}\) , by Lemma 7 and Minkowski inequality, we get \begin{eqnarray*} \|T_{\Omega}^{b}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})} &\leq& \int_{B_{j}}\left\|\left|\frac{\Omega(\cdot-y)}{|\cdot-y|^{n}}-\frac{\Omega(\cdot)}{|\cdot|^{n}} \right|(b(\cdot)-b(y))\chi_{k} \right\|_{L^{p(\cdot)}(\mathbb{R}^{n})}|g_{j}(y)|\mathrm{d}y\\ &\leq&\int_{B_{j}}\left\|\left|\frac{\Omega(\cdot-y)}{|\cdot-y|^{n}}-\frac{\Omega(\cdot)}{|\cdot|^{n}} \right||b(\cdot)-b_{B_{j}}|\chi_{k} \right\|_{L^{p(\cdot)}(\mathbb{R}^{n})}|g_{j}(y)|\mathrm{d}y\\ &&+\int_{B_{j}}\left\|\left|\frac{\Omega(\cdot-y)}{|\cdot-y|^{n}}-\frac{\Omega(\cdot)}{|\cdot|^{n}} \right|\chi_{k}\right\|_{L^{p(\cdot)}(\mathbb{R}^{n})}|b_{B_{j}}-b(y)||g_{j}(y)|\mathrm{d}y. \end{eqnarray*} Since \( \tilde{p}(\cdot)>1 \text{and} 1/p(\cdot)=1/\tilde{p}(\cdot)+1/r\) . Since \( r>p^{+}\) , so by Lemma 4 and Lemma 11, we get \begin{eqnarray*} \|T_{\Omega}^{b}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})} &\leq& \left\|\left|\frac{\Omega(\cdot-y)}{|\cdot-y|^{n}}-\frac{\Omega(\cdot)}{|\cdot|^{n}}\right| \right\|_{L^{r}(\mathbb{R}^{n})}\left\|b(\cdot)-b_{B_{j}}\chi_{k} \right\|_{L^{\tilde{p}(\cdot)}(\mathbb{R}^{n})}\int_{B_{j}}|g_{j}(y)|dy\\ &&+\left\|\left|\frac{\Omega(\cdot-y)}{|\cdot-y|^{n}}-\frac{\Omega(\cdot)}{|\cdot|^{n}} \right|\right\|_{L^{r}(\mathbb{R}^{n})} \|\chi_{B_{k}}\|_{L^{\tilde{p}(\cdot)}(\mathbb{R}^{n})}\int_{B_{j}}|b_{B_{j}}-b(y)||g_{j}(y)|dy \end{eqnarray*} From (8) and Lemmas 4-6, we get

Therefore, when \( 0< q\leq1\) , we obtain \begin{eqnarray*} &E'_{2}&=\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}| \|T_{\Omega}^{b}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\nonumber\\ &&\leq C \|b\|^{q}_{\ast}\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}| (k-j)2^{(j-k)(\beta+n\delta_{2})-j\alpha(0)}\right)^{q}\nonumber\\ &&\leq C \|b\|^{q}_{\ast}\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}| (k-j) 2^{[(j-k)(\beta+n\delta_{2})-\alpha(0)]}\right)^{q}\nonumber\\ &&\leq C \|b\|^{q}_{\ast}\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{j=-\infty}^{L}|\lambda_{j}|^{q}\left(\sum\limits_{k=j+1}^{-1} (k-j) 2^{[(j-k)(\beta+n\delta_{2})-\alpha(0)]}\right)^{q}\nonumber\\ &&\leq C \|b\|^{q}_{\ast}\Upsilon. \end{eqnarray*} When \( 0< q\leq\infty,\) and \( 1/q+1/q^{\prime}=1,\) by \( n\delta_{2}\leq \alpha(0) < \beta +n\delta_{2}\) and Hölder inequality, we have \begin{eqnarray*} &E'_{2}&=\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}2^{k\alpha(0)q}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}| \|T_{\Omega}^{b}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\nonumber\\ &&\leq C \|b\|^{q}_{\ast}\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{k=-\infty}^{L}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}|^{q} 2^{[(j-k)(\beta+n\delta_{2})-\alpha(0)]\frac{q}{2}}\right)\nonumber\\ && \times\left(\sum\limits_{j=-\infty}^{k-1} (k-j)^{q^{\prime}} 2^{[(j-k)(\beta+n\delta_{2})-\alpha(0)]\frac{q^{\prime}}{2}}\right)^{\frac{q}{q^{\prime}}}\nonumber\\ &&\leq C \|b\|^{q}_{\ast}\sup\limits_{L\leq0,L\in \mathbb{Z}} 2^{-L\lambda q}\sum\limits_{j=-\infty}^{L}|\lambda_{j}|^{q}\sum\limits_{k=j+1}^{-1} 2^{[(j-k)(\beta+n\delta_{2})-\alpha(0)]\frac{q}{2}}\nonumber\\ &&\leq C \|b\|^{q}_{\ast}\Upsilon. \end{eqnarray*} Now we prove that \( F'\leq C\Upsilon.\) \begin{eqnarray*} F'&=& \sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\left\|T_{\Omega}^{b}(h)\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})}\leq \sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\left(\sum\limits_{j=k}^{\infty}|\lambda_{j}| \|T_{\Omega}^{b}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\\ && + \sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}| \|T_{\Omega}^{b}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\\ &:=&F'_{1}+F'_{2}. \end{eqnarray*} To estimate \( F'_{1}\) . By the boundedness of the \( T^{b}_{\Omega}\) on \( L^{p(\cdot)}(\mathbb{R}^{n})\) (see[13]) and following the same way as we estimated \( F_{1}\) in Theorem 13, we get For \( F_{2}\) , when \( 0< q\leq1\) , by inequality (18) and \( n\delta_{2}\leq \alpha(0)< \varepsilon+n\delta_{2}\) , we have when \( 0< q\leq\infty,\) and \( 1/q+1/q^{\prime}=1.\) by \( n\delta_{2}\leq \alpha(0) < \beta+n\delta_{2}\) and Hölder inequality, we obtain \begin{eqnarray*} F'_{2}&=&\sum\limits_{k=-\infty}^{-1}2^{k\alpha(0)q}\left( \sum\limits_{j=-\infty}^{k-1}|\lambda_{j}| \|T_{\Omega}^{b}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\nonumber\\ &\leq& C \|b\|^{q}_{\ast}\sum\limits_{k=-\infty}^{-1}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}|^{q} 2^{[(j-k)(\beta+n\delta_{2})-\alpha(0)]\frac{q}{2}}\right)\nonumber \times\left(\sum\limits_{j=-\infty}^{k-1} (k-j)^{q^{\prime}} 2^{[(j-k)(\beta+n\delta_{2})-\alpha(0)]\frac{q^{\prime}}{2}}\right)^{\frac{q}{q^{\prime}}}\nonumber\\ &\leq& C \|b\|^{q}_{\ast}\sum\limits_{j=-\infty}^{-1}|\lambda_{j}|^{q}\sum\limits_{k=j+1}^{-1} 2^{[(j-k)(\beta+n\delta_{2})-\alpha(0)]\frac{q}{2}}\nonumber\\ &\leq& C \|b\|^{q}_{\ast}\Upsilon. \end{eqnarray*} Finally, we show that \( G'\leq C\Upsilon.\) \begin{eqnarray*} G'&=&\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q} \left\|T_{\Omega}^{b}(h)\chi_{k}\right\|^{q}_{L^{p(\cdot)}(\mathbb{R}^{n})},\\ &\leq& \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q}\left(\sum\limits_{j=k}^{\infty}|\lambda_{j}| \|T_{\Omega}^{b}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\\ && +\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}| \|T_{\Omega}^{b}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q},\\ &:=&G'_{1}+G'_{2}. \end{eqnarray*} For \( G'_{1}\) , by the boundedness of the commutator \( T_{\Omega}^{b}\) in \( L^{p(\cdot)}(\mathbb{R}^{n})\) (see [13] ), and following the same way as we estimated \( G_{1}\) in Theorem 13, we get Now, we estimate \( G'_{2}\) . When \( 0< q< \infty,\) and \( 1/q+1/q^{\prime}=1,\) from inequality (18), since \( n\delta_{2} \leq\alpha(0), \alpha_{\infty}< \beta+n\delta_{2}\) and applying Hölder inequality, we obtain \begin{eqnarray*} G'_{2}&=&\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}| \|T_{\Omega}^{b}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\right)^{q}\\ &\leq& C \|b\|_{\ast}\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q}\left(\sum\limits_{j=-\infty}^{k-1}|\lambda_{j}| (k-j) 2^{(j-k)(\beta+n\delta_{2})-j\alpha_{j}}\right)^{q}\\ &\leq& C \|b\|_{\ast}\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q}\left(\sum\limits_{j=-\infty}^{-1}|\lambda_{j}| (k-j) 2^{(j-k)(\beta+n\delta_{2})-j\alpha(0)}\right)^{q}\\ && + C \|b\|_{\ast}\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{k\alpha_{\infty}q}\left(\sum\limits_{j=0}^{k-1}|\lambda_{j}| (k-j) 2^{(j-k)(\beta+n\delta_{2})-j\alpha_{\infty}}\right)^{q}\\ &\leq& C \|b\|_{\ast}\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{[\alpha_{\infty}-(\beta+n\delta_{2})]kq}\left(\sum\limits_{j=-\infty}^{-1}|\lambda_{j}| (k-j) 2^{[(\beta+n\delta_{2})-\alpha(0)]j}\right)^{q}\\ && + C \|b\|_{\ast}\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}\left(\sum\limits_{j=0}^{k-1}|\lambda_{j}| (k-j) 2^{(j-k)[(\beta+n\delta_{2})-\alpha_{\infty}]}\right)^{q}\\ &\leq& C \|b\|_{\ast}\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}2^{[\alpha_{\infty}-(\beta+n\delta_{2})]kq}\left(\sum\limits_{j=-\infty}^{-1}|\lambda_{j}| 2^{j\frac{[(\beta+n\delta_{2})-\alpha(0)]}{2}}\right)^{q}\left(\sum\limits_{j=-\infty}^{-1}(k-j)^{q^{\prime}} 2^{j[(\beta+n\delta_{2})-\alpha(0)]\frac{q^{\prime}}{2}}\right)^{\frac{q}{q^{\prime}}}\\ && + C \|b\|_{\ast}\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}\left(\sum\limits_{j=0}^{k-1}|\lambda_{j}| 2^{(j-k)\frac{[(\beta+n\delta_{2})-\alpha_{\infty}]}{2}}\right)^{q} \left( \sum\limits_{j=0}^{k-1}(k-j)^{q^{\prime}}2^{(j-k)[(\beta+n\delta_{2})-\alpha_{\infty}] \frac{q^{\prime}}{2}}\right)^{\frac{q}{q^{\prime}}}\nonumber\\ &\leq& C \|b\|_{\ast}\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\left(\sum\limits_{j=-\infty}^{-1}|\lambda_{j}|^{q} 2^{[(\beta+n\delta_{2})-\alpha(0)]j\frac{q}{2}}\right)+ C \|b\|_{\ast}\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}\sum\limits_{j=0}^{k-1}|\lambda_{j}|^{q} 2^{(j-k)[(\beta+n\delta_{2})-\alpha_{\infty}]\frac{q}{2}}\nonumber\\ &\leq& C \|b\|_{\ast}\left[\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{j=-\infty}^{-1}|\lambda_{j}|^{q}\right.\left.+ \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{k=0}^{L}\sum\limits_{j=0}^{k-1}|\lambda_{j}|^{q} 2^{(j-k)[(\beta+n\delta_{2})-\alpha_{\infty}]\frac{q}{2}}\right]\nonumber\\ &\leq& C \|b\|_{\ast}\left[\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{j=-\infty}^{-1}|\lambda_{j}|^{q}\right. \left.+ \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{j=0}^{L-1}|\lambda_{j}|^{q}\sum\limits_{k=j+1}^{L} 2^{(j-k)[(\beta+n\delta_{2})-\alpha_{\infty}]\frac{q}{2}}\right]\nonumber\\ &\leq& C \|b\|_{\ast}\left[\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{j=-\infty}^{-1}|\lambda_{j}|^{q} + \sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{j=0}^{L-1}|\lambda_{j}|^{q}\right]\nonumber\\ &\leq& C \|b\|_{\ast}\sup\limits_{L>0,L\in \mathbb{Z}}2^{-L\lambda q}\sum\limits_{j=-\infty}^{L-1}|\lambda_{j}|^{q}\nonumber\\ &\leq& C \|b\|_{\ast}\Upsilon. \end{eqnarray*} The proof is completed.

Theorem 15. Suppose that \(b\in \dot{\Lambda}_{\gamma}(\mathbb{R}^{n})(0< \gamma\leq 1), p_{1}(\cdot), p_{2}(\cdot)\in \mathcal{B}(\mathbb{R}^{n})\) be such that \( p_{1}^{+}< n/\gamma, 1/p_{1}(x)-1/p_{2}(x)=\gamma/n, \Omega\in L^{r}(\mathbb{S}^{n-1})(r>q^{+}_{2})\) with \( 1\leq r' < p_{1}^{-}\) and satisfies $$ \int^{1}_{0}\frac{w_{r}(\delta)}{\delta^{1+\gamma}}d\delta < \infty.$$ Let \( 0< q< \infty, 0\leq\lambda< \infty \text{and} \alpha(\cdot)\in L^{\infty}(\mathbb{R}^{n})\) satisfies conditions (4) and (5) of Proposition 3. If \( 2\lambda\leq\alpha(\cdot), n\delta_{2}\leq \alpha(0),\alpha_{\infty}< \gamma+n\delta_{2},\) then \( [b,T_{\Omega}]\) is bounded from \( HM\dot{K}^{\alpha(\cdot),q}_{p_{1}(\cdot),\lambda}\) or \( \left(HMK^{\alpha(\cdot),q}_{p_{1}(\cdot),\lambda}\right)\) to \( M\dot{K}^{\alpha(\cdot),q}_{p_{2}(\cdot),\lambda}\) or \( \left(MK^{\alpha(\cdot),q}_{p_{2}(\cdot),\lambda}\right).\)

Proof. The prove of this Theorem follows almost similarly to that of Theorem 14. Instead of giving all details, we only give the modifications required for the estimation of \( E'', F'' \text{and} G''\) . Note that if \( x\in B_{k} \text{for each} k\in\mathbb{Z}, y\in B_{j}\) and \( j\leq k-1\) . Let \( \tilde{p}(\cdot)>1 \text{and} 1/p(\cdot)=1/\tilde{p}(\cdot)+1/r\) , since \( r>p^{+}\) , so by Lemmas 10 and 12, we get \begin{eqnarray*} \|T_{\Omega}^{b}(g_{j})\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}&\leq& \int_{B_{j}}\left\|\left|\frac{\Omega(\cdot-y)}{|\cdot-y|^{n}}-\frac{\Omega(\cdot)}{|\cdot|^{n}} \right|(b(\cdot)-b(y))\chi_{k}\right\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}|g_{j}(y)|dy\\ &\leq& \int_{B_{j}}\left\|\left|\frac{\Omega(\cdot-y)}{|\cdot-y|^{n}}-\frac{\Omega(\cdot)}{|\cdot|^{n}} \right||b(\cdot)-b_{B_{j}}|\chi_{k}\right\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}|g_{j}(y)|dy\\ && +\int_{B_{j}}\left\|\left|\frac{\Omega(\cdot-y)}{|\cdot-y|^{n}}-\frac{\Omega(\cdot)}{|\cdot|^{n}}\right|\chi_{k} \right\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}|b_{B_{j}}-b(y)||g_{j}(y)|dy \end{eqnarray*} Since \( \tilde{p_{2}}(\cdot)>1 \text{and} 1/p_{2}(\cdot)=1/\tilde{p_{2}}(\cdot)+1/r\) , by \( r>p^{+}\) and Lemmas 11 and 12, we deduced \begin{eqnarray*} \|T_{\Omega}^{b}(g_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}&\leq& \left\|\left|\frac{\Omega(\cdot-y)}{|\cdot-y|^{n}}-\frac{\Omega(\cdot)}{|\cdot|^{n}}\right| \right\|_{L^{r}(\mathbb{R}^{n})}\left\|(b-b_{B_{j}})\chi_{k} \right\|_{L^{\tilde{p}_{2}(\cdot)}(\mathbb{R}^{n})}\int_{B_{j}}|g_{j}(y)|dy\\ && +\left\|\left|\frac{\Omega(\cdot-y)}{|\cdot-y|^{n}}-\frac{\Omega(\cdot)}{|\cdot|^{n}}\right|\right\|_{L^{r}(\mathbb{R}^{n})} \|\chi_{B_{k}}\|_{L^{\tilde{p}_{2}(\cdot)}(\mathbb{R}^{n})}\int_{B_{j}}|b_{B_{j}}-b(y)||g_{j}(y)|dy\\ &\leq& C \left\|\left|\frac{\Omega(\cdot-y)}{|\cdot-y|^{n}}-\frac{\Omega(\cdot)}{|\cdot|^{n}}\right| \right\|_{L^{r}(\mathbb{R}^{n})}\left\|(b-b_{B_{j}})\chi_{k} \right\|_{L^{\tilde{p}_{2}(\cdot)}(\mathbb{R}^{n})}\|g_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{j}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})}\\ && +\left\|\left|\frac{\Omega(\cdot-y)}{|\cdot-y|^{n}}-\frac{\Omega(\cdot)}{|\cdot|^{n}}\right| \right\|_{L^{r}(\mathbb{R}^{n})}\|\chi_{B_{k}}\|_{L^{\tilde{p}_{2}(\cdot)}(\mathbb{R}^{n})} \|(b-b_{B_{j}})\chi_{B_{j}}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})} \|g_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}. \end{eqnarray*} By Lemma 9, we have

by Lemma 4-6,we have From this, following the same calculations as we did for \( E', F' \text{and} G'\) in Theorem 14, we get

Acknowledgments

This paper is supported by Shendi University.

Authorcontributions

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

Competing Interests

The author(s) do not have any competing interests in the manuscript.