Commutators of fractional integral with variable kernel on variable exponent Herz-Morrey spaces

1. Introduction

Muckenhoupt and Wheeden [1] have proved the boundedness of the fractional integral operators \(T_{\Omega,\mu}\) with power weights from \(L^{p}\) to \(L^{q}\). The boundedness of the fractional integral operators were studied by Calderón and Zygmund [2]. Ding, Chen and Fan [3] introduced the properties of \(T_{\Omega,\mu}\) on Hardy spaces. The theory of the variable exponent function spaces has been rapidly developed after the work [4] where Kováčik and Rákosník proved fundamental properties of Lebesgue spaces with variable exponent. After that, many researchers work in this direction, see [5, 6, 7, 8, 9, 10, 11]. Izuki [7] defined the class of Herz-Morrey spaces with variable exponent and considered the boundedness of the fractional integral on these spaces. In [11] the author's studied the boundedness of the fractional integral with variable kernel on \(M\dot{K}_{q,p(\cdot)}^{\alpha, \lambda}(\mathbb{R}^{n})\) spaces.

Let \( S^{n-1} (n\geq 2) \) be the unit sphere in \(\mathbb{R}^{n}\) with normalized Lebesgue measure \( d\sigma(x')\). A function \(\Omega(x,z)\) defined on \( \mathbb{R}^{n} \times \mathbb{R}^{n}\) is said to be in \({L}^{\infty}(\mathbb{R}^{n})\times{L^{r}(S^{n-1})}(r\geq{1})\), if \(\Omega\) satisfies the following two conditions:
(i) For any \( x,z \in{\mathbb{R}^{n}}\) and any \(\lambda > 0\), one has \(\Omega{(x,\lambda{z})}=\Omega(x,z)\);
(ii) \(\|\Omega\|_{L^{\infty}(\mathbb{R}^{n})\times{L^{r}(S^{n-1})}}:=\sup\limits_{x\in\mathbb{R}^{n}}\left(\int_{s^{n-1}}|{\Omega(x ,z')}|^{r}\mathrm{d}{\sigma}(z')\right)^{\frac{1}{r}}{< \infty}.\)
For \( 0\leq \mu < n \) the fractional integral operator with variable kernel \(T_{\Omega,\mu}\) is defined by $$T_{\Omega,\mu}f (x)= \int_{\mathbb{R}^{n}}\frac{\Omega(x, x-y )}{|x -y|^{n-\mu}}f(y)\mathrm{d}y.$$ The commutators of the fractional integral is defined by $$ [b^{m},T_{\Omega,\mu}]f (x)= \int_{\mathbb{R}^{n}}\frac{\Omega(x, x-y )}{|x -y|^{n-\mu}}(b(x)- b(y))^{m}f(y)\mathrm{d}y. $$ Throughout this paper, Let \(E\) be a Lebesgue measurable set in \(\mathbb{R}^{n}\) with measure \(|E|>0\), \(\chi_{E}\) means its characteristic function. Now, introduce the definition of the variable exponent Lebesgue spaces.

Definition [5] Let \(p(\cdot): E \rightarrow {[1,\infty)}\) be a measurable function, the Lebesgue space with variable exponent \(L^{p(\cdot)}(E)\) is defined by $$ L^{p(\cdot)}(E)= \{{ f~ \mbox{is measurable}: \int_{E}\left(\frac{|f(x)|}{\eta}\right)^{p(x)} \mathrm{d}x <\infty}~ \mbox{for some constant } \eta > 0\}. $$

The space \(L _{loc}^{p(\cdot)} {(E)}\) is defined by $$ L_{loc}^{p(\cdot)} {(E)}= \{ \mbox {f is measurable}: f\in {L^{p(\cdot)} {(K)}}~\mbox{for all compact}~K\subset E\}. $$ The Lebesgue spaces \(L^{p(\cdot)} {(E)}\) is a Banach spaces with the norm defined by $$ \|f\|_{L^{p(\cdot)}(E)}= \inf\left\{\eta> 0 : \int_{E}\left(\frac{|f(x)|}{\eta}\right)^{p(x)}\mathrm{d}x \leq 1\right\}. $$

We denote \(p_{-}=\) ess inf \(\{p(x): x \in E\} , \) \( p_{+}=\) ess sup\( \{p(x): x \in E\} \). Then \(\mathcal{P}(E)\) consists of all \(p(\cdot)\) satisfying \(p_{-} > 1\) and \(p_{+} < \infty\). 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 [7] Let \(\alpha \in\mathbb{R}\), \(0< q < \infty\), \(p(\cdot)\in \mathcal{P} (\mathbb{R}^{n})\) and \(0 \leq\lambda< \infty\). The homogeneous Herz\(-\) Morrey spaces with variable exponent \(M\dot{K}_{q,p(\cdot)}^{\alpha,\lambda}(\mathbb{R}^{n})\) is defined by $$ M\dot{K}_{q,p(\cdot)}^{\alpha, \lambda}(\mathbb{R}^{n})= \left\{f\in {L_{Loc}^{p(\cdot)}}(\mathbb{R}^{n}\backslash\{0\}) : \|f\|_{M\dot{K}_{q,p(\cdot)}^{\alpha, \lambda}(\mathbb{R}^{n})}< \infty \right\}, $$ where $$ \|f\|_{M\dot{K}_{q,p{(\cdot)}}^{\alpha,\lambda}(\mathbb{R}^{n})}:= \sup_{L\in \mathbb{Z}} 2^{-L\lambda} \left\{\sum_{K=-\infty}^{L} 2^{k\alpha q}\|f\chi_{k}\|_{L^{p(\cdot)}}^{q}\right\}^{1/q}. $$

2. Properties of variable Lebesgue spaces

In this section, we state some properties of variable exponent Lebesgue spaces.

Proposition 3. Assume that \(p(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\) satisfies the follows inequalities: \begin{align*} | p(x) - p(y)| &\leq \frac{ -C}{Log( |x - y|)},\qquad | x - y| \leq 1/ 2;\\ | p(x) - p(y)| &\leq \frac{ C}{Log( e +|x|)},\qquad |y|\geq|x|; \end{align*} then, we have \(p(\cdot)\in \mathfrak{B}(\mathbb{R}^{n})\).

Lemma 4 [4] Let \(p(\cdot): \mathbb{R}^{n} \rightarrow [1 , \infty)\), for all function \(f\) and \(g\), there exists the fact $$ \int_{\mathbb{R}^{n}}|f(x) g(x)| dx \leq C\|f\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\|g\|_{L^{p'(\cdot)}(\mathbb{R}^{n})}. $$

Lemma 5. [5] Assume that \(E\) is a Lebesgue measurable subset of \(\mathbb{R}^{n}\) with positive measure and \( p(\cdot)\in \mathcal{P}\), if \(f: E\times E \longrightarrow \mathbb{R}\) be a measurable function (with respect to product measure) such that for almost every \(y\in E\), \(f(\cdot,y)\in L^{p(\cdot)}(E)\). Then we have $$ \left\|\int_{E}f(\cdot,y)dy\right\|_{L^{p(\cdot)}(E)}\leq C \int_{E}\left\|f(\cdot,y)\right\|_{L^{p(\cdot)}(E)}\mathrm{d}y. $$

Lemma 6. 12 If \(0 < \mu < n \), and \(\Omega\in{L}^{\infty}(\mathbb{R}^{n})\times{L^{r}(S^{n-1})}(r\geq1)\) satisfies the \(L^{r}-\) Dini condition. If there exists an \( 0< \alpha < 1/2\) such that \( |y|< \alpha_{0}R ,\) we have $$ \left(\int_{R< |x|< 2 R} \left|\frac{\Omega(x, x- y)}{|x - y|^{n - \mu})}- \frac{\Omega(x,x)}{|x|^{n-\mu}}\right|^{r} \mathrm{d}x \right)^{{\frac{1}{r}}}\leq CR^{(\frac{n}{r}- n +\mu)} \left(\frac{|y|}{R} + \int_{|y|/2R}^{|y|/R} \frac{\omega_{r}(\delta)}{\delta} \mathrm{d}\delta \right). $$

Lemma 7. [13] If \(x \in \mathbb{R}^{n}\) and defined \(\widetilde{q}(x)\) by \(\frac{1}{p(x)}= \frac{1}{q} + \frac{1}{\widetilde{q}(x)}\), for all measurable function \(f\) and \(g\), we have $$ \|f(x)g(x)\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\leq C \|g(x)\|_{L^{r}(\mathbb{R}^{n})}\|f(x)\|_{L^{\widetilde{q}(\cdot)}(\mathbb{R}^{n})}. $$

Lemma 8. [13] Suppose that \(p(\cdot) \in \mathfrak{B}(\mathbb{R}^{n})\) and \( 0 < p^{-}\leq p^{+} < \infty,\) (1)\ For any cube (or ball) and \(|Q|\leq 2^{n}\), all the \(\chi \in Q\), then: \(\|\chi_{Q}\|_{L^{p(\cdot)}}\approx |Q|^{1/p(x)}\). (2)\ For any cube (or ball) and \(|Q|\geq 1\), then \(\|\chi_{Q}\|_{L^{p(\cdot)}}\approx |Q|^{1/p_{\infty}}\), where \( p_{\infty} = \lim_{ x \rightarrow\infty} p(x)\).

Lemma 9. [8] If \(p(\cdot) \in \mathfrak{B}(\mathbb{R}^{n})\), then there exist constants \(C, \delta,\delta_{1} > 0 \) such that for all balls \(B\) in \(\mathbb{R}^{n}\) and all measurable subset \(S\subset B\) $$ \frac{\|\chi_{S}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}} {\|\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}} \leq C \left( \frac{|S|}{|B|}\right)^{\delta}, \frac{\|\chi_{S}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})}} {\|\chi_{B}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})}} \leq C \left( \frac{|S|}{|B|}\right)^{\delta_{1}}. $$

Lemma 10. [8] If \(p(\cdot) \in \mathfrak{B}(\mathbb{R}^{n})\), there exists constant \(C > 0\) such that for any balls B in \(\mathbb{R}^{n}\), we have $$ \frac{1}{|B|}\|\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})} \|\chi_{B}\|_{L^{p'(\cdot)}(\mathbb{R}^{n})}\leq C. $$

Lemma 11. [14] Assume that \(b^{m} \in BMO(\mathbb{R}^{n})\), \(p_{1}(\cdot)\in \mathfrak{B}(\mathbb{R}^{n})\), \(\Omega\in{L}^{\infty}(\mathbb{R}^{n})\times{L^{r}(S^{n-1})}\). Let \(0< \gamma\leq\frac{n}{( p_{1})_{+}} \), \(\frac{1}{p_{1}(x)} - \frac{1}{p_{2}(x)} = \frac{\gamma}{n}\), then for all \(f \in L^{p_{1}(\cdot)}(\mathbb{R}^{n})\), we have $$ \|T^{b^{m}}_{\Omega , \gamma} f\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\leq C\|b\|^{m}_{BMO(\mathbb{R}^{n})} \| f\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}. $$

Lemma 12. [15] Assume that \(b^{m}\in Lip_{\beta}(\mathbb{R}^{n})\), \(p_{1}(\cdot)\in \mathfrak{B}(\mathbb{R}^{n})\), \(\Omega\in{L}^{\infty}(\mathbb{R}^{n})\times{L^{r}(S^{n-1})}\). Let \(0< \gamma\leq\frac{n}{( p_{1})_{+}},\) \(\frac{1}{p_{1}(x)} - \frac{1}{p_{2}(x)} = \frac{\gamma+m\beta}{n}\), then for all \(f \in L^{p_{1}(\cdot)}(\mathbb{R}^{n})\) we have $$ \|T^{b^{m}}_{\Omega , \gamma} f\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\leq C \|b\|^{m}_{Lip_{\beta}} \| f\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}. $$

Lemma 13. [10] Let \(b^{m}\in Lip_{\beta}(\mathbb{R}^{n})\); \(m\) is a positive integer, and there exist constants \(C> 0\), such that for any \( k,j \in\mathbb{Z}\) with \(k>j\), we have (1)\ \(C^{-1}\|b\|^{m}_{Lip_{\beta}}\leq |B|^{-m\beta/n}\|\chi_{B}\|^{-1}_{L^{p(\cdot)}(\mathbb{R}^{n})}\|( b- b_{B})^{m}\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})} \leq C \|b\|^{m}_{Lip_{\beta}}\); (2)\ \(\|( b- b_{B_{j}})^{m}\chi_{B_{k}}\|_{L^{p(\cdot)}(\mathbb{R}^{n})} \leq C |B_{k}|^{m\beta/n} \|b\|^{m}_{Lip_{\beta}}\| \chi_{B_{k}}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}.\)

Lemma 14. [16] Let \(b^{m} \in BMO(\mathbb{R}^{n})\); \(m\) is a positive integer, and there exist constants \(C> 0\), such that for any \( k,j \in\mathbb{Z}\) with \(k>j\), we have (1)\ \( C^{-1}\|b\|^{m}_{*}\leq \sup\limits_{B} \frac{1}{\|\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}}\|( b- b_{B})^{m}\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})} \leq C\|b\|_{*}^{m};\) (2)\ \(\|( b- b_{B_{j}})^{m}\chi_{B_{k}}\|_{L^{p(\cdot)}(\mathbb{R}^{n})} \leq C (k - j)^{m}\|b\|^{m}_{*}\| \chi_{B_{k}}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}.\)

3. Main theorems and their proofs

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

Theorem 15. Assume that \(b^{m} \in BMO(\mathbb{R}^{n})\), \( 0< \mu< n, 0< \beta\leq 1, \lambda < \alpha < n\delta_{1} + \beta , 0< q_{1} \leq q_{2}< \infty.\) Let \(\Omega\in{L}^{\infty}(\mathbb{R}^{n})\times{L^{r}(S^{n-1})}(r > p_{2}^{+})\), and the integral modulus of continuity \(\omega_{r}(\delta)\) satisfying

If \(p_{1}(\cdot) \in \mathfrak{B}(\mathbb{R}^{n})\) such that \(0< \mu\leq\frac{n}{( p_{1})_{+}}\), \(\frac{1}{p_{1}(x)} - \frac{1}{p_{2}(x)} = \frac{\mu}{n}\). Then for all \(f \in {M\dot{K}_{q_{1},p_{1}(\cdot)}^{\alpha,\lambda}(\mathbb{R}^{n})}\), we have $$\|T^{b^{m}}_{\Omega , \mu} f\|_{M\dot{K}_{q_{2},p_{2}(\cdot)}^{\alpha,\lambda}(\mathbb{R}^{n})} \leq C\|b\|^{m}_{*} \| f\|_{M\dot{K}_{q_{1},p_{1}(\cdot)}^{\alpha,\lambda}(\mathbb{R}^{n})}.$$

Proof. If \(f \in {M\dot{K}_{q_{1},p_{1}(\cdot)}^{\alpha,\lambda}(\mathbb{R}^{n})}\) arbitrarily, we use the following inequality $$\left(\sum\limits_{k=1}^{\infty}a_{k}\right)^{q}\leq \sum\limits_{k=1}^{\infty}a_{k}^{q} ~~~~~~~~~~ \mbox{such that}~~~~ (a_{1}, a_{2}.....) \geq 0,$$ we have \begin{align*} \|T^{b^{m}}_{\Omega , \mu} f\|^{q_{1}}_{M\dot{K}_{q_{2},p_{2}(\cdot)}^{\alpha,\lambda}(\mathbb{R}^{n})}&= \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}} \left\{\sum_{K=-\infty}^{L} 2^{k\alpha q_{2}}\|T^{b^{m}}_{\Omega ,\mu} (f)\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}^{q_{2}}\right\}^{q_{1}/q_{2}}\\ &\leq \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}} \left\{\sum_{K=-\infty}^{L} 2^{k\alpha q_{1}}\|T^{b^{m}}_{\Omega ,\mu} (f)\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}^{q_{1}}\right\}. \end{align*} Let \( f(x) = \sum\limits_{k=-\infty}^{\infty}f(x)\chi_{k}= {\sum\limits_{k=- \infty}^{ \infty}} f_{j}(x)\). Then we have \begin{align*} \|T^{b^{m}}_{\Omega , \mu} f\|^{q_{1}}_{M\dot{K}_{q_{2},p_{2}(\cdot)}^{\alpha,\lambda}(\mathbb{R}^{n})} &\leq \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}} \sum\limits_{k=-\infty}^{L} 2^{k\alpha q_{1}} \left( \sum\limits_{j=-\infty}^{\infty}\|T^{b^{m}}_{\Omega ,\mu} (f_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \right)^{q_{1}}\\ &\leq \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}}\sum_{k=-\infty}^{L} 2^{k\alpha q_{1}} \left( \sum\limits_{j=-\infty}^{k-2}\|T^{b^{m}}_{\Omega ,\mu} (f_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \right)^{q_{1}}\\ &~~~~+ \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}}\sum\limits_{k=-\infty}^{L} 2^{k\alpha q_{1}} \left( \sum\limits_{j=k-1}^{\infty}\|T^{b^{m}}_{\Omega ,\mu} (f_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \right)^{q_{1}}\\ &= U_{1}+ U_{2}. \end{align*} First, we consider \( U_{1}\). By the vanishing condition of \(f_{j}\), applying Lemma 5 and Minkowski inequality when \(j\leq k-2\) we have \begin{eqnarray*} \quad\|T^{b^{m}}_{\Omega , \mu} (f_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} &\leq& \int_{B_{j}} |f_{j}(y)|\left\| {\left|\frac{\Omega(x , x- y)}{|x -y|^{n - \mu}} - \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right| }|b(x)-b(y)|^{m}\chi_{k} \right\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \mathrm{d}y \\ &\leq& \int_{B_{j}} |f_{j}(y)|\left\| {\left|\frac{\Omega(x , x- y)}{|x -y|^{n - \mu}} - \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right| }|b(x)-b_{B}|^{m}\chi_{k} \right\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \mathrm{d}y\\ &&+ \int_{B_{j}} |b_{B}-b(y)|^{m}|f_{j}(y)|\left\| {\left|\frac{\Omega(x , x- y)}{|x -y|^{n - \mu}} - \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right| }\chi_{k} \right\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \mathrm{d}y\\ &=& U_{11} + U_{12}. \end{eqnarray*} For \(U_{11}\), we define \( \frac{1}{p_{2}(x)}= \frac{1}{r} + \frac{1}{\widetilde{p}_{2}(x)},\) such that \({\widetilde{p_{2}}(x)}> 1\), by Lemma 7 and Lemma 14, we have \begin{align*} &~~\left\| { \left|\frac{\Omega(x , x-y)}{|x -y|^{n - \mu}}- \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right|} |b(x)-b_{B}|^{m} \chi_{k} \right\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\\ &\leq \left\|\frac{\Omega(x , x-y)}{|x -y|^{n - \mu}} - \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right\|_{L^{r}(\mathbb{R}^{n})} \||b(x)-b_{B}|^{m}\chi_{k}\|_{L^{\widetilde{p_{2}}(x)}(\mathbb{R}^{n})}\\ &\leq\left\|\frac{\Omega(x , x-y)}{|x -y|^{n - \mu}} - \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right\|_{L^{r}(\mathbb{R}^{n})} \||b(x)-b_{B_{j}}|^{m}\chi_{B_{k}}\|_{L^{\widetilde{p_{2}}(x)}(\mathbb{R}^{n})}\\ &\leq\left\|\frac{\Omega(x , x-y)}{|x -y|^{n - \mu}} - \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right\|_{L^{r}(\mathbb{R}^{n})} \||b(x)-b_{B_{j}}|^{m}\chi_{B_{k}}\|_{L^{\widetilde{p_{2}}(x)}(\mathbb{R}^{n})}\\ &\leq\left\|\frac{\Omega(x , x-y)}{|x -y|^{n - \mu}} - \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right\|_{L^{r}(\mathbb{R}^{n})} (k-j)^{m}\|b\|^{m}_{*} \|\chi_{B_{k}}\|_{L^{\widetilde{p_{2}}(\cdot)}(\mathbb{R}^{n})}. \end{align*} According to Lemma 8 and the formula \( \frac{1}{\tilde{p}_{2}(x)} = \frac{1}{p_{2}(x)} - \frac{1}{r}\), we have $$\|\chi_{B_{k}}\|_{L^{\widetilde{p_{2}}(x)}(\mathbb{R}^{n})} \approx \|\chi_{B_{k}}\|_{L^{p_{1}(x)}(\mathbb{R}^{n})} |B|^{\frac{-1}{r} - \frac{\mu}{n}}.$$ Applying Lemma 6, noting that \(2^{j -k }\leq 2 ^{(j - k)\beta}\) we get \begin{align*} \left\|\frac{\Omega(x , x-y)}{|x -y|^{n - \mu}} - \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right\|_{L^{r}(\mathbb{R}^{n})} &\leq CR^{(\frac{n}{r}- n +\mu)} \left(\frac{|y|}{2^{k-1}} + \int\limits_{|y|/2^{k}}^{|y|/2^{k-1}} \frac{\omega_{r}(\delta)}{\delta} \mathrm{d}\delta \right) \\ &\leq CR^{(\frac{n}{r}- n +\mu)} \left( 2^{j-k} + 2^{(j-k)\beta} \int\limits_{0}^{1} \frac{\omega_{r}(\delta)}{\delta^{1+\beta}} \mathrm{d}\delta\right) \\ &\leq CR^{(\frac{n}{r}- n +\mu)} 2^{(j-k)\beta} \left( 1+ \int\limits_{0}^{1} \frac{\omega_{r}(\delta)}{\delta^{1+\beta}} \mathrm{d}\delta \right)\\ &=C2^{{(k-1)}{(\frac{n}{r}- n +\mu)}} 2^{(j-k)\beta}. \end{align*} Thus, we have \begin{align*} U_{11}&\leq C(k-j)^{m}\|b\|^{m}_{*} 2^{-kn +(j-k)\beta}\|\chi_{B_{k}}\|_{L^{p_{1}(x)}(\mathbb{R}^{n})} \int_{B_{j}}|f_{j}(y)| \mathrm{d}y\\ &\leq C(k-j)^{m}\|b\|^{m}_{*} 2^{-kn +(j-k)\beta} \|f\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{j}}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})}. \end{align*} For \(U_{12}\), similar to \(U_{11}\), we have \begin{eqnarray*} \left\| { \left|\frac{\Omega(x , x-y)}{|x -y|^{n - \mu}}- \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right|} \chi_{k} \right\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} &\leq& \left\|\frac{\Omega(x , x-y)}{|x -y|^{n - \mu}} - \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right\|_{L^{r}(\mathbb{R}^{n})}\|\chi_{k}\|_{L^{\widetilde{p_{2}}(x)}(\mathbb{R}^{n})}\\ &\leq& C 2^{-kn +(j-k)\beta} \|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}. \end{eqnarray*} By Lemma 14, we obtain \begin{align*} U_{12} &\leq C 2^{-kn +(j-k)\beta} \|f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\||b_{B_{j}}-b(y)|^{m}\chi_{B_{j}}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\\ &\leq C \|b\|^{m}_{*} 2^{-kn +(j-k)\beta} \|f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\|\chi_{B_{j}}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})}. \end{align*} So, $$\|T^{b^{m}}_{\Omega , \mu} (f_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \leq C (k-j)^{m}\|b\|^{m}_{*} 2^{-kn +(j-k)\beta} \|f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\|\chi_{B_{j}}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})}. $$ Using Lemma 4, Lemma 9 and Lemma 10, we have \begin{eqnarray*} \|T^{b^{m}}_{\Omega , \mu} (f_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} &\leq& C (k-j)^{m}\|b\|^{m}_{*} 2^{-kn +(k-j)\beta} \| f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{j}}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\\ &\leq& C(k-j)^{m}\|b\|^{m}_{*} 2^{(j-k)\beta} \| f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{j}}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})} 2^{-kn} \|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\\ &\leq& C (k-j)^{m}\|b\|^{m}_{*}2^{(j-k)\beta} \|f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\frac{\|\chi_{B_{j}}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})}} {\|\chi_{B_{k}}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})}}\\ &\leq& C(k-j)^{m}\|b\|^{m}_{*} 2^{(j-k)\beta}2^{(j- k)n\delta_{1}}\| f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\\ &\leq& C(k-j)^{m}\|b\|^{m}_{*} 2^{(j-k)(\beta+n\delta_{1})} \| f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}. \end{eqnarray*} Then we have \begin{align*} U_{1}&\leq C \|b\|^{m q_{1}}_{*} \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}}\sum_{k=-\infty}^{L} \left( \sum\limits_{j=-\infty}^{k-2} (k-j)^{m}2^{\alpha{k}}2^{(j-k)(\beta+n\delta_{1})}\| f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\right)^{q_{1}}\\ &\leq C \|b\|^{m q_{1}}_{*} \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}}\sum_{k=-\infty}^{L} \left( \sum\limits_{j=-\infty}^{k-2} (k-j)^{m}2^{\alpha{j}}2^{(j-k)(\beta+n\delta_{1}-\alpha)} \| f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \right)^{q_{1}} \end{align*} When \(1< q_{1}< \infty\), take \(1/q_{1}+1/q'_{1}=1\). Noting that \( \alpha < n\delta_{1} + \beta \), by the Hölder's inequality we obtain

When \(0< q_{1}\leq1\), we have Finally, we estimate \(U_{2}\). By the boundedness of the \(T^{b^{m}}_{\Omega,\mu}\) on \(L^{p(\cdot)}(\mathbb{R}^{n})\)(Lemma 11 ), we have \begin{align*} U_{2}&\leq C \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}}\sum_{k=-\infty}^{L} 2^{k\alpha q_{1}} \left( \sum_{j=k-1}^{\infty}\|T^{b^{m}}_{\Omega ,\mu} (f_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \right)^{q_{1}}\\ &\leq C\|b\|^{m q_{1}}_{*} \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}}\sum_{k=-\infty}^{L} 2^{k\alpha q_{1}} \left(\sum\limits_{j=k-1}^{\infty}\| f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \right)^{q_{1}}\\ &\leq C\|b\|^{m q_{1}}_{*} \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}}\sum_{k=-\infty}^{L} \left( \sum\limits_{j=k-1}^{\infty} 2^{\alpha j} 2^{(k- j)\alpha } \| f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\right)^{q_{1}}\\ &\leq C\|b\|^{m q_{1}}_{*} \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}}\sum_{k=-\infty}^{L} \left(\sum\limits_{j=k-1}^{k+ 1} 2^{\alpha j} 2^{(k- j)\alpha } \| f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\right)^{q_{1}}\\ &~~~~~+ \|b\|^{m q_{1}}_{*} \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}}\sum_{k=-\infty}^{L} \left(\sum\limits_{j=k+2}^{\infty} 2^{\alpha j} 2^{(k- j)\alpha } \| f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\right)^{q_{1}}\\ &= U_{21} + U_{22}. \end{align*} First, we consider \(U_{21}\), then we have \begin{align*} U_{21} &\leq C\|b\|^{m q_{1}}_{*}\sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}}\sum_{k=-\infty}^{L} \left(\sum\limits_{j=k-1}^{k+ 1} 2^{\alpha j} 2^{(k- j)\alpha } \| f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \right)^{q_{1}}\end{align*} For \(U_{22}\), when \(\lambda < \alpha \) we have Thus, by (2)-(5), we finishes the proof of Theorem 15.

Theorem 16. Assume that \(b^{m} \in Lip_{\beta}(\mathbb{R}^{n})\), \( 0< \mu< n, 0< \beta\leq 1, \lambda < \alpha < n\delta_{1} + \beta , 0< q_{1} \leq q_{2}< \infty.\) Let \(\Omega\in{L}^{\infty}(\mathbb{R}^{n})\times{L^{r}(S^{n-1})}(r > p_{2}^{+})\), and the integral modulus of continuity \(\omega_{r}(\delta)\) satisfying 1. If \(p_{1}(\cdot) \in \mathfrak{B}(\mathbb{R}^{n})\) such that \(0< \mu+m\beta\leq\frac{n}{( p_{1})_{+}}\), \(\frac{1}{p_{1}(x)} - \frac{1}{p_{2}(x)} = \frac{\mu+m\beta}{n}\). Then for all \(f \in {M\dot{K}_{q_{1},p_{1}(\cdot)}^{\alpha,\lambda}(\mathbb{R}^{n})}\), we have $$\|T^{b^{m}}_{\Omega , \mu} f\|_{M\dot{K}_{q_{2},p_{2}(\cdot)}^{\alpha,\lambda}(\mathbb{R}^{n})} \leq C\|b\|^{m}_{Lip_{\beta}} \| f\|_{M\dot{K}_{q_{1},p_{1}(\cdot)}^{\alpha,\lambda}(\mathbb{R}^{n})}.$$

Proof. If \(f \in {M\dot{K}_{q_{1},p_{1}(\cdot)}^{\alpha,\lambda}(\mathbb{R}^{n})}\) arbitrarily, we use the following inequality $$\left(\sum\limits_{k=1}^{\infty}a_{k}\right)^{q}\leq \sum\limits_{k=1}^{\infty}a_{k}^{q} ~~~~~~~~~~ \mbox{such that}~~~~ (a_{1}, a_{2}.....) \geq 0,$$ we have \begin{align*} \|T^{b^{m}}_{\Omega , \mu} f\|^{q_{1}}_{M\dot{K}_{q_{2},p_{2}(\cdot)}^{\alpha,\lambda}(\mathbb{R}^{n})}&= \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}} \left\{\sum_{K=-\infty}^{L} 2^{k\alpha q_{2}}\|T^{b^{m}}_{\Omega ,\mu} (f)\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}^{q_{2}}\right\}^{q_{1}/q_{2}}\\ &\leq \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}} \left\{\sum_{K=-\infty}^{L} 2^{k\alpha q_{1}}\|T^{b^{m}}_{\Omega ,\mu} (f)\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}^{q_{1}}\right\}. \end{align*} Let \( f(x) = \sum_{k=-\infty}^{\infty}f(x)\chi_{k}= {\sum_{k=- \infty}^{ \infty}} f_{j}(x)\). Then we have \begin{align*} \|T^{b^{m}}_{\Omega , \mu} f\|^{q_{1}}_{M\dot{K}_{q_{2},p_{2}(\cdot)}^{\alpha,\lambda}(\mathbb{R}^{n})} &\leq \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}} \sum\limits_{k=-\infty}^{L} 2^{k\alpha q_{1}} \left( \sum\limits_{j=-\infty}^{\infty}\|T^{b^{m}}_{\Omega ,\mu} (f_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \right)^{q_{1}}\\ &\leq \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}}\sum_{k=-\infty}^{L} 2^{k\alpha q_{1}} \left( \sum\limits_{j=-\infty}^{k-2}\|T^{b^{m}}_{\Omega ,\mu} (f_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \right)^{q_{1}}\\ &~~~~+ \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}}\sum\limits_{k=-\infty}^{L} 2^{k\alpha q_{1}} \left( \sum\limits_{j=k-1}^{\infty}\|T^{b^{m}}_{\Omega ,\mu} (f_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \right)^{q_{1}}\\ &= A_{1}+ A_{2}. \end{align*} First, we consider \( A_{1}\). By the vanishing condition of \(f_{j}\) and Lemma 5, the Minkowski inequality when \(j\leq k-2\) we have \begin{eqnarray*} \|T^{b^{m}}_{\Omega , \mu} (f_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} &\leq&\int_{B_{j}} |f_{j}(y)|\left\| {\left|\frac{\Omega(x , x- y)}{|x -y|^{n - \mu}} - \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right| }|b(x)-b(y)|^{m}\chi_{k} \right\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \mathrm{d}y \\ &\leq& \int_{B_{j}} |f_{j}(y)|\left\| {\left|\frac{\Omega(x , x- y)}{|x -y|^{n - \mu}} - \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right| }|b(x)-b_{B}|^{m}\chi_{k} \right\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \mathrm{d}y\\ &&+ \int_{B_{j}} |b_{B}-b(y)|^{m}|f_{j}(y)|\left\| {\left|\frac{\Omega(x , x- y)}{|x -y|^{n - \mu}} - \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right| }\chi_{k} \right\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \mathrm{d}y\\ &=& A_{11} + A_{12}. \end{eqnarray*} For \(A_{11}\), we define \( \frac{1}{p_{2}(x)}= \frac{1}{r} + \frac{1}{\widetilde{p}_{2}(x)}\) such that \({\widetilde{p_{2}}(x)}> 1\), by Lemma 7 and Lemma 13, we have \begin{align*} &~~\left\| { \left|\frac{\Omega(x , x-y)}{|x -y|^{n - \mu}}- \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right|} |b(x)-b_{B}|^{m} \chi_{k} \right\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\\ &\leq \left\|\frac{\Omega(x , x-y)}{|x -y|^{n - \mu}} - \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right\|_{L^{r}(\mathbb{R}^{n})} \||b(x)-b_{B}|^{m}\chi_{k}\|_{L^{\widetilde{p_{2}}(x)}(\mathbb{R}^{n})}\\ &\leq\left\|\frac{\Omega(x , x-y)}{|x -y|^{n - \mu}} - \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right\|_{L^{r}(\mathbb{R}^{n})} \||b(x)-b_{B_{j}}|^{m}\chi_{B_{k}}\|_{L^{\widetilde{p_{2}}(x)}(\mathbb{R}^{n})}\\ &\leq\left\|\frac{\Omega(x , x-y)}{|x -y|^{n - \mu}} - \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right\|_{L^{r}(\mathbb{R}^{n})} \||b(x)-b_{B_{j}}|^{m}\chi_{B_{k}}\|_{L^{\widetilde{p_{2}}(x)}(\mathbb{R}^{n})}\\ &\leq\left\|\frac{\Omega(x , x-y)}{|x -y|^{n - \mu}} - \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right\|_{L^{r}(\mathbb{R}^{n})} |B_{k}|^{m\beta/n}\|b\|^{m}_{Lip_{\beta}(\mathbb{R}^{n})} \|\chi_{B_{k}}\|_{L^{\widetilde{p_{2}}(\cdot)}(\mathbb{R}^{n})}. \end{align*} According to Lemma 8 and the formula \( \frac{1}{\tilde{p}_{2}(x)} = \frac{1}{p_{2}(x)} - \frac{1}{r}\), we have $$\|\chi_{B_{k}}\|_{L^{\widetilde{p_{2}}(x)}(\mathbb{R}^{n})} \approx \|\chi_{B_{k}}\|_{L^{p_{1}(x)}(\mathbb{R}^{n})} |B|^{\frac{-1}{r} - \frac{(\mu+m\beta)}{n}}.$$ By Lemma 7 know that $$\left\|\frac{\Omega(x , x-y)}{|x -y|^{n - \mu}} - \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right\|_{L^{r}(\mathbb{R}^{n})}\leq C2^{{(k-1)}{(\frac{n}{r}- n +\mu)}} 2^{(j-k)\beta}.$$ Thus, we have \begin{align*} A_{11}&\leq C |B_{k}|^{m\beta/n}\|b\|^{m}_{Lip_{\beta}(\mathbb{R}^{n})}2^{{(k-1)}{(\frac{n}{r}- n +\mu)}} 2^{(j-k)\beta} \|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} |B|^{\frac{-1}{r} - \frac{(\mu+m\beta)}{n}} \int_{B_{j}} |f_{j}(y)|\mathrm{d}y\\ &\leq C \|b\|^{m}_{Lip_{\beta}(\mathbb{R}^{n})} 2^{-kn +(j-k)\beta}\|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \int_{B_{j}}|f_{j}(y)| \mathrm{d}y.\\ &\leq C \|b\|^{m}_{Lip_{\beta}(\mathbb{R}^{n})} 2^{-kn +(j-k)\beta} \|f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{j}}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})}\|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}. \end{align*} For \(A_{12}\), we obtain that \begin{eqnarray*} \left\| { \left|\frac{\Omega(x, x-y)}{|x -y|^{n - \mu}}- \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right|} \chi_{k} \right\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} &\leq& \left\|\frac{\Omega(x , x-y)}{|x -y|^{n - \mu}} - \frac{\Omega(x ,x)}{|x|^{n -\mu}}\right\|_{L^{r}(\mathbb{R}^{n})}\|\chi_{k}\|_{L^{\widetilde{p_{2}}(\cdot)}(\mathbb{R}^{n})}\\ &\leq& C |B_{k}|^{-m\beta/n}2^{-kn +(j-k)\beta} \|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \end{eqnarray*} Then, by Lemma 13, we get \begin{align*} A_{12}&\leq C |B_{k}|^{-m\beta/n}2^{-kn +(j-k)\beta} \|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \|f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\||b_{B_{j}}-b(y)|^{m}\chi_{B_{j}}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})}\end{align*}\begin{align*} &\leq C\|b\|^{m}_{Lip(\mathbb{R}^{n})} \frac{|B_{j}|^{m\beta/n}}{|B_{k}|^{m\beta/n}} 2^{-kn+(j-k)\beta}\|f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\|\chi_{B_{j}}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\\ &\leq C\|b\|^{m}_{Lip_{\beta}(\mathbb{R}^{n})} 2^{-kn+(j-k)\beta}\|f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\|\chi_{B_{j}}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}. \end{align*} So, we have that $$\|T^{b^{m}}_{\Omega , \mu} (f_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \leq C 2^{-kn +(j-k)\beta} \|f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\|\chi_{B_{j}}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})}. $$ Using Lemma 4, Lemma 9 and Lemma 10, we have \begin{eqnarray*} \|T^{b^{m}}_{\Omega , \mu} (f_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} &\leq& C \|b\|^{m}_{Lip_{\beta}(\mathbb{R}^{n})} 2^{-kn +(k-j)\beta} \| f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{j}}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{k}}\|_{L^{p_{1}(x)}(\mathbb{R}^{n})}\\ &\leq& C\|b\|^{m}_{Lip_{\beta}(\mathbb{R}^{n})} 2^{(j-k)\beta} \| f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{j}}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})} 2^{-kn} \|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\\ &\leq& C \|b\|^{m}_{Lip_{\beta}(\mathbb{R}^{n})}2^{(j-k)\beta} \|f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\frac{\|\chi_{B_{j}}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})}} {\|\chi_{B_{k}}\|_{L^{p'_{1}(\cdot)}(\mathbb{R}^{n})}}\\ &\leq& \|b\|^{m}_{Lip_{\beta}(\mathbb{R}^{n})}2^{(j-k)\beta}2^{(j- k)n\delta_{1}}\| f_{j}\|_{L^{p_{1}(x)}(\mathbb{R}^{n})}\\ &\leq& \|b\|^{m}_{Lip_{\beta}(\mathbb{R}^{n})} 2^{(j-k)(\beta+n\delta_{1})} \| f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}. \end{eqnarray*} When \(1< q_{1}< \infty\), take \(1/q_{1}+1/q'_{1}=1\). Noting that \( \alpha < n\delta_{1} + \beta \), by the Hölder's inequality we have

When \(0< q_{1}\leq1\), we have Finally, we estimate \(A_{2}\). By the boundedness of the \(T^{b^{m}}_{\Omega,\mu}\) on \(L^{p(\cdot)}(\mathbb{R}^{n})\)(Lemma 12 ), we have \begin{align*} A_{2}&\leq C \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}}\sum_{k=-\infty}^{L} 2^{k\alpha q_{1}} \left( \sum_{j=k-1}^{\infty}\|T^{b^{m}}_{\Omega ,\mu} (f_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \right)^{q_{1}}\\ &\leq C\|b\|^{m q_{1}}_{Lip_{\beta}(\mathbb{R}^{n})} \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}}\sum_{k=-\infty}^{L} 2^{k\alpha q_{1}} \left(\sum\limits_{j=k-1}^{\infty}\| f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \right)^{q_{1}}\\ &\leq C\|b\|^{m q_{1}}_{Lip_{\beta}(\mathbb{R}^{n})} \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}}\sum_{k=-\infty}^{L} \left( \sum\limits_{j=k-1}^{\infty} 2^{\alpha j} 2^{(k- j)\alpha } \| f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\right)^{q_{1}}\\ &\leq C\|b\|^{m q_{1}}_{Lip_{\beta}(\mathbb{R}^{n})} \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}}\sum_{k=-\infty}^{L} \left(\sum\limits_{j=k-1}^{k+ 1} 2^{\alpha j} 2^{(k- j)\alpha } \| f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\right)^{q_{1}}\\ &+ \|b\|^{m q_{1}}_{Lip_{\beta}(\mathbb{R}^{n})} \sup_{L\in \mathbb{Z}} 2^{-L\lambda {q_{1}}}\sum_{k=-\infty}^{L} \left(\sum\limits_{j=k+2}^{\infty} 2^{\alpha j} 2^{(k- j)\alpha } \| f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\right)^{q_{1}}.\\ \end{align*} The rest of the proof is the same as the proof of \(U_{2}\) in Theorem 15, we omit the details there. Then, we can easily see that By (6)-(8) the proof of Theorem 16 is complete.

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.