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 $$ \rho_{p}(h):=\int_{\Omega}\left(\frac{|h(x)|}{\mu}\right)^{p(x)}dx 0.$$ 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 \( p(\cdot)\in \mathcal{P}(\mathbb{R}^{n}), 0< q< \infty, 0\leq \lambda n+1\) . 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 5. 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 (4) and (5) of Proposition (3), 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 \( 1< q1 \text{and} 1/p(\cdot)=1/\tilde{p}(\cdot)+1/r\) . 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 \( p_{1}^{+}q^{+}_{2})\) 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.

Author Contributions

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.