Multilinear fractional integral with rough kernel on variable exponent Morrey-Herz spaces

1. Introduction

Suppose that \( S^{n-1} (n> 1) \) denote the unit sphere in \(\mathbb{R}^{n}\) with the normalized Lebesgue measure \( \mathrm{d}\sigma(x’)\). Let \(\Omega\in{L^{s}(S^{n-1})}(1< s< \infty)\) be a homogeneous function of degree zero on \(\mathbb{R}^{n}.\) The multilinear fractional integral operator with rough kernel \(T^{A}_{\Omega,\mu}( 0< \mu< n)\) is defined by $$ T^{A}_{\Omega,\mu}f (x)= \int_{\mathbb{R}^{n}} \frac{\Omega( x-y )}{|x -y|^{n-\mu+m}}R_{m+1}(A;x,y)f(y)dy, $$ where \(A\) is a function defined on \(\mathbb{R}^{n}\) and \(R_{m+1}(A;x,y)\) is the \(mth\) reminder of Taylor series of \(A\) at \(x\) about \(y\). More precisely $$ R_{m+1}(A;x,y)= A(x) -\sum_{|\gamma|\leq m}\frac{1}{\gamma!}D^{\gamma} A(y)(x-y)^{\gamma}, $$ when \(m=1\), \(T^{A}_{\Omega,\mu}\) is just the commutators of the fractional integral with rough kernel with function \(A.\) $$ T^{A}_{\Omega,\mu}f (x)= \int_{ \mathbb{R}^{n}}\frac{\Omega( x-y )}{|x -y|^{n-\mu}}(A(x)-A(y))f(y)dy. $$ The multilinear fractional maximal operator with rough kernel is defined as $$ M^{A}_{\Omega,\mu}f (x) = \sup\limits_{r>0}\frac{1}{r^{(n-\mu + m)}}\int\limits_{|x-y|< r} |\Omega(x-y)R_{m+1}(A;x,y)f(y)|dy. $$

In 1975, Coifman and Meyer [1] introduced multilinear integral and the boundedness of the multilinear fractional integral operator on Lebesgue spaces established in [2, 3, 4]. Li and Tao [5] discussed the boundedness of multilinear commutators with rough kernels on Morrey-Herz spaces.

On the other hand, the theory of the variable exponent function spaces has been rapidly developed after it was introduced by Kováčik and Rákosník [6]. After that, many researchers work in this direction has been done, see for example [7, 8, 9, 10, 11, 12, 13]. The boundedness of the multilinear fractional integral operator on variable Lebesgue spaces are established in [14, 15]. Recently, Lu and Zhu [16] established the boundedness of the multilinear Calderón-Zygmund singular operators on Morrey-Herz spaces with variable exponents.

In this article, we study the boundedness of the multilinear fractional integral operator and multilinear fractional maximal operator with rough kernels on variable exponent Lebesgue spaces. The boundedness of the multilinear fractional integral operator is established on Morrey-Herz spaces \({MK}_{q(\cdot),p(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})\).

Throughout this paper, let \(|E|\) be a Lebesgue measurable set in \(\mathbb{R}^{n}\) with measure \(|E|>0\) and \(\chi_{E}\) be its characteristic function. We shall recall some definitions.

Definition 1. [7]. Let \(p(\cdot): E \rightarrow {[1,\infty)}\) be a measurable function, the variable exponent Lebesgue spaces \(L^{p(\cdot)}(E)\) is defined as $$ L^{p(\cdot)}(E)= \{{ f~ \mbox{is measurable}: \int_{E}\left(\frac{|f(x)|}{\eta}\right)^{p(x)} dx < \infty}~ \mbox{for some constant } \eta > 0\}. $$ The space \(L _{loc}^{p(\cdot)} {(E)}\) is defined as $$ L_{loc}^{p(\cdot)} {(E)}= \{ \mbox {f is measurable} : f\in {L^{p(\cdot)} {(K)}}~for~all~compact~K\subset E\}. $$ The relation between Lebesgue spaces \(L^{p(\cdot)} {(E)}\) and Banach spaces is defined as $$ \|f\|_{L^{p(\cdot)}(E)}= \inf\left\{\eta> 0: \int_{E}\left(\frac{|f(x)|}{\eta}\right)^{p(x)}dx \leq 1\right\}. $$

We denote \(p_{-}=\) essinf \(\{p(x): x \in E\} , \) \( p_{+}=\) esssup\( \{p(x): x \in E\} \). Then \(\mathcal{P}(E)\) consists of all \(p(\cdot)\) satisfying \(p_{-} > 1\) and \(p_{+} < \infty\). Next, we give the definition of Morrey-Herz space with variable exponents \( q(\cdot),p(\cdot),\alpha(\cdot)\). Let \(B_{k}=\{ x\in\mathbb{R}^{n}: |x|\leq 2^{k}\} , C_{k}= B_{k}\backslash B_{k-1} , \chi_{k}= \chi_{C_{k}} , \) \( k \in{\mathbb{Z}}\).

Definition 2. [18]. Let \(q (\cdot),p(\cdot)\in \mathcal{P} (\mathbb{R}^{n})\), \( 0\leq \lambda < \infty\) and \( \alpha(\cdot): \mathbb{R}^{n} \longrightarrow \mathbb{R}\) with \(\alpha \in L^{\infty}(\mathbb{R}^{n})\). The nonhomogeneous Morrey-Herz space with variable exponents \({MK}_{q(\cdot),p(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})\) is defined as $$ {MK}_{q(\cdot),p(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})= \{f\in {L_{\mathrm{loc}}^{p(\cdot)}}(\mathbb{R}^{n}\backslash\{0\}) : \|f\|_{{MK}_{q(\cdot),p(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})}< \infty \}, $$ where $$ \|f\|_{{MK}_{q(\cdot),p(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})} = \inf\left\{ \eta> 0 : \sup\limits_{k_{0}\in z } 2^{-k_{0}\lambda} \sum\limits_{k=0}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}|f\chi_{k}|}{\eta}\right)^{q(\cdot)}\right\|_{L^{\frac{p(\cdot)}{q(\cdot)}}}\leq 1\right\}. $$ The homogeneous Morrey-Herz space with variable exponents \(M\dot{K}_{q(\cdot),p(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})\) is defined as $$ M\dot{K}_{q(\cdot),p(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})= \{f\in {L_{\mathrm{loc}}^{p(\cdot)}}(\mathbb{R}^{n}\backslash\{0\}) : \|f\|_{M\dot{K}_{q(\cdot),p(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})}< \infty \}, $$ where $$ \|f\|_{M\dot{K}_{q(\cdot),p(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})} = \inf\left\{ \eta> 0 : \sup\limits_{k_{0}\in z} 2^{-k_{0}\lambda} \sum\limits_{k=-\infty}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}|f\chi_{k}|}{\eta}\right)^{q(\cdot)}\right\|_{L^{\frac{p(\cdot)}{q(\cdot)}}}\leq 1\right\}. $$

2. Preliminaries

In this section, we give some properties of variable exponents that will be helpful in proving our main results.

Proposition 1.[7]. If \(p(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\) satisfies the follows inequalities: \begin{align*} | p(x) – p(y)|&\leq \frac{ -C}{log( |x – y|)}~,| x – y| \leq 1/ 2;\\ | p(x) – p(y)|&\leq \frac{ C}{log( e +|x|)}~,|y|\geq|x|. \end{align*} then, we have \(p(\cdot)\in \mathfrak{B}(\mathbb{R}^{n})\).

Proposition 2. [14]. Suppose that \(p_{1}(\cdot)\in \mathfrak{B}(\mathbb{R}^{n})\), \(\Omega\in{L^{r}(S^{n-1})}\), \(0 < \mu\leq\frac{n}{( p_{1})_{+}}\), \(\frac{1}{p_{1}(x)}- \frac{1}{p_{2}(x)} = \frac{\mu}{n}\), then for all \(f \in L^{p_{1}(\cdot)}(\mathbb{R}^{n})\), we have $$ \|M_{\Omega,\mu} f\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\leq C\| f\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}. $$

Now, we recall some lemmas.

Lemma 1. [7].

1. Let \(p(\cdot): \mathbb{R}^{n} \rightarrow [ 1, \infty)\), for all function \(f\) and \(g\), then $$ \int_{\mathbb{R}^{n}}|f(x) g(x)| \mathrm{d}x \leq C\|f\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\|g\|_{L^{p'(\cdot)}(\mathbb{R}^{n})}. $$
2. If \( p(\cdot), q (\cdot), r(\cdot) \in \mathbb{R}^{n} \), \( p(\cdot)\) and \( \frac{1}{p(\cdot)}= \frac{1}{q(\cdot)} + \frac{1}{r(\cdot)}\). Then there exists a constant C such that for all \( f \in L^{q(\cdot)}(\mathbb{R}^{n}), g \in L^{r(\cdot)}(\mathbb{R}^{n}) \), we have $$ \|fg\|_{L^{p(\cdot)}}\leq C\|f\|_{L^{q(\cdot)}(\mathbb{R}^{n})}\|g\|_{L^{r(\cdot)}(\mathbb{R}^{n})}. $$

Lemma 2.[7]. Let \(x \in \mathbb{R}^{n}\) and \(\frac{1}{p(\cdot)}= \frac{1}{q} + \frac{1}{\widetilde{q}(\cdot)}\), then for all measurable function \(f\) and \(g\), we have $$ \|f(x)g(x)\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\leq C \|g(x)\|_{L^{q}(\mathbb{R}^{n})}\|f(x)\|_{L^{\widetilde{q}(\cdot)}(\mathbb{R}^{n})}. $$

Lemma 3.[9]. Suppose that \(p(\cdot) \in \mathfrak{B}(\mathbb{R}^{n})\) and \( 0 < p^{-}\leq p^{+} < \infty \), then we have

1. for any cube \(|Q|\leq 2^{n}\), and all the \(\chi \in Q\), we have \(\|\chi_{Q}\|_{L^{p(\cdot)}}\approx |Q|^{1/p(x)}\),
2. for any cube \(|Q|\geq 1\), we have \(\|\chi_{Q}\|_{L^{p(\cdot)}}\approx |Q|^{1/p_{\infty}}\), where \( p_{\infty} = \lim_{ x \rightarrow\infty} p(x)\).

Lemma 4.[11]. If \(p(\cdot) \in \mathfrak{B}(\mathbb{R}^{n})\), then there exist constants \(\delta_{1},\delta_{2},C > 0 \) such that for all balls \(B\) in \(\mathbb{R}^{n}\) and all measurable subset \(S\subset B\), we have $$ \frac{\|\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}} {\|\chi_{S}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}} \leq C \frac{|B|}{|S|},~ \frac{\|\chi_{S}\|_{L^{p’_{1}(\cdot)}(\mathbb{R}^{n})}} {\|\chi_{B}\|_{L^{p’_{1}(\cdot)}(\mathbb{R}^{n})}} \leq C \left(% \begin{array}{ccc} \frac{|S|}{|B|}\end{array}\right)^{\delta_{1}}, ~ \frac{\|\chi_{S}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}} {\|\chi_{B}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}} \leq C \left(% \begin{array}{ccc} \frac{|S|}{|B|}\end{array}\right)^{\delta_{2}}. $$

Lemma 5.[19]. For \(p(\cdot) \in \mathfrak{B}(\mathbb{R}^{n})\), 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 6.[13]. Let \(p(\cdot), q(\cdot) \in\mathcal{P}(\mathbb{R}^{n})\). If \(f\in L^{p(\cdot)q(\cdot)}\), then $$ \min ( \|f\|_{L^{p(\cdot)q(\cdot)}}^{q_{+}}~,~ \|f\|_{L^{p(\cdot)q(\cdot)}}^{q_{-}} )\leq\||f|^{q(\cdot)}\|_{L^{p(\cdot)}}\leq\max \left( \|f\|_{L^{p(\cdot)q(\cdot)}}^{q_{+}} ~,~ \|f\|_{L^{p(\cdot)q(\cdot)}}^{q_{-}} \right). $$

Lemma 7.[20]. Let \(b\in BMO (\mathbb{R}^{n})\), where \(n\) is a positive integer, and let the constant \(C> 0\). Then for any \( l,j \in\mathbb{Z}\) with \(l>j\), we have

1. \( C^{-1}\|b\|^{n}_{*}\leq \sup\limits_{B} \frac{1}{\|\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}}\|( b- b_{B})^{n}\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})} \leq C\|b\|_{*}^{n},\)
2. \(\|( b- b_{B_{j}})^{n}\chi_{B_{k}}\|_{L^{p(\cdot)}(\mathbb{R}^{n})} \leq C (l – j)^{n}\|b\|^{n}_{*}\| \chi_{B_{k}}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}.\)

Lemma 8.[2]. For any \( \varepsilon > 0\) with \( 0< \mu -\varepsilon< \mu +\varepsilon< n\), we have $$ | T^{A} _{\Omega,\mu} f(x)|\leq C [M^{A}_{\Omega,\mu+\varepsilon}f(x)]^\frac{1}{2}[M^{A}_{\Omega,\mu-\varepsilon}f(x)]^\frac{1}{2}.$$

Lemma 9.[4]. Let \(A\) be a function with derivatives of order \(m\) in \( \dot{\wedge}_{\beta}( 0< \beta 0 \) such that $$ |R_{m+1}(A;x,y)|\leq C \left(\sum\limits_{|\gamma|=m}\|D^{\gamma}A\|_{\dot{\wedge}_{\beta}}\right)|x-y|^{m+\beta}. $$

Lemma 10.[21]. Let \(b(x)\) be a function on \(\mathbb{R}^{n}\) and \( D^{\gamma}b \in L^{q}_{loc}(\mathbb{R}^{n})\), where \(q>n\), then $$ |R_{m}(b,x,y)|\leq C |x-y|^{m}\sum\limits_{|\gamma|=m}\left(\frac{1}{B(x,y)}\int\limits_{B(x,y)}|D^{\gamma}b(z)|dz\right)^{\frac{1}{q}}. $$ where \(B(x,y)\) is the cube centered at \(x\) and having diameter \(5\sqrt{n}|x-y|\).

Lemma 11.[7]. Let \(q(\cdot) \in\mathcal{P}(\mathbb{R}^{n})\) and \( q^{+}0\), we have $$ \||f|^{s}\|_{q(\cdot)(\mathbb{R}^{n})}=\|f\|^{s}_{s q(\cdot)(\mathbb{R}^{n})}. $$

Lemma 12. Let \({D^{\gamma}A}\in BMO(\mathbb{R}^{n})(|\gamma|=|m|, m\geq1),\) \(\Omega\in{L^{r}(S^{n-1})}\), \(p(\cdot) \in \mathfrak{B}(\mathbb{R}^{n})\) and \(\frac{1}{p_{1}(x)} – \frac{1}{p_{2}(x)} = \frac{\mu}{n}\), then we have $$ \|M_{\Omega,\mu}^{A}f(x)\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\leq C \sum_{|\gamma|=m}\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})} \|f \|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}, $$ where C is independent of \(f\) and \(A.\)

Proof. If \( \widetilde{M}^{A}_{\Omega,\mu}\) is defined by: $$ \widetilde{M}^{A}_{\Omega,\mu}f(x)= \sup\limits_{r>0}\frac{1}{r^{(n-\mu + m)}}\int\limits_{\frac{r}{2}< |x-y|< r} |\Omega(x-y)R_{m+1}(A;x,y)f(y)|\mathrm{d}y. $$ Let \(Q(x,r)\) be the cube centered at \( x \) and having diameter \(5\sqrt{nr}\). If \(\frac{r}{2}< |x-y|< r \), by Lemma 10, we have \begin{align*} |R_{m+1}(A;x,y)| &= |R_{m+1}(A_{k};x,y)|\leq |R_{m}(A_{k};x,y)| +\sum_{|\gamma|=m}\frac{1}{\gamma!}|D^{\gamma}A_{k}(y)||x-y|^{m}\\ &\quad\leq C |x-y|^{m}\left(\sum_{|\gamma|=m}\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})} +\sum_{|\gamma|=m} |D^{\gamma} A(y)-m_{B_{k}}(D^{\gamma}A)| \right). \end{align*} By using Hölder’s inequality, we get \begin{align*} \widetilde{M}^{A}_{\Omega,\mu}f(x)&= \sup\limits_{r>0}\frac{1}{r^{n-\mu }}\int\limits_{\frac{r}{2}< |x-y|< r}\left(\sum_{|\gamma|=m}\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})} +\sum_{|\gamma|=m} |D^{\gamma} A(y)-m_{B_{k}}(D^{\gamma}A)| \right)\\ &\quad\times|\Omega(x-y)f(y)|\mathrm{dy} \\ &\leq C\sum_{|\gamma|=m}\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})} \left(M_{|\Omega|^{t},\mu t}(|f|^{t})(x)\right)^{\frac{1}{t}}. \end{align*} By the boundedness of the fractional maximal operator on \(L^{p(\cdot)}(\mathbb{R}^{n})\) spaces, we obtain that \begin{align*} \|\widetilde{M}^{A}_{\Omega,\mu}f(x)\|_{{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}}&\leq C \sum_{|\gamma|=m}\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})} \|M_{|\Omega|^{t},\mu t}(|f|^{t})(x)\|^{\frac{1}{t}}_{L^{\frac{p_{2}(\cdot)}{t}}}\\ &\leq C\sum_{|\gamma|=m}\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})} \||f|^{t}\|^{\frac{1}{t}}_{L^{\frac{p_{1}(\cdot)}{t}}}\\ &\leq C\sum_{|\gamma|=m}\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})} \|f\|_{L^{{p_{1}(\cdot)}}(\mathbb{R}^{n})}. \end{align*} Since \({M}^{A}_{\Omega,\mu}f(x)\leq \widetilde{M}^{A}_{\Omega,\mu}f(x)\) for all \(x\in\mathbb{R}^{n}\), we have \begin{align*} \| {M}^{A}_{\Omega,\mu}f(x)\|_{{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}} &\leq \| \widetilde{M}^{A}_{\Omega,\mu}f(x)\|_{{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}}\\ &\leq C \sum_{|\gamma|=m}\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})} \|f\|_{L^{{p_{1}(\cdot)}}(\mathbb{R}^{n})}. \end{align*} Then, we get $$ \|M_{\Omega,\mu}^{A}f(x)\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\leq C \sum_{|\gamma|=m}\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})} \|f \|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}. $$ This completes the proof of Lemma 12.

Lemma 13. Let \({D^{\gamma}A}\in BMO(\mathbb{R}^{n})(|\gamma|=|m|, m\geq1)\), \(\Omega\in{L^{r}(S^{n-1})}\), \( p(\cdot) \in \mathfrak{B}(\mathbb{R}^{n})\) and \(\frac{1}{p_{1}(x)} – \frac{1}{p_{2}(x)} = \frac{\mu}{n}\), then we have $$ \|T_{\Omega,\mu}^{A}f(x)\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\leq C \sum_{|\gamma|=m}\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})} \|f \|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}. $$ where C is independent of \(f\) and \(A.\)

Proof. Let \( 0< \varepsilon < \min (\mu , n-\mu)\), and \(r(\cdot): \mathbb{R}^{n} \longrightarrow [1, +\infty )\), and let \begin{align*} &\frac{1}{p_{1}(\cdot)} – \frac{1}{\frac{r(\cdot)p_{2}(\cdot)}{2}} = \frac{\mu -\varepsilon}{2},\\ &\frac{1}{p_{1}(\cdot)} – \frac{1}{\frac{r'(\cdot)p_{2}(\cdot)}{2}} = \frac{\mu +\varepsilon}{2}. \end{align*} By Lemma 8 and applying Hölder's inequality, we have $$ \|T^{A} _{\Omega,\mu} f(x)\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\leq C\|(M^{A}_{\Omega,\mu+\varepsilon}f)^{\frac{1}{2}}\|_{L^{p_{2}(\cdot)r'(\cdot)}} \|(M^{A}_{\Omega,\mu-\varepsilon}f)^{\frac{1}{2}}\|_{L^{p_{2}(\cdot)r(\cdot)}}. $$ Since \begin{align*} \|(M^{A}_{\Omega,\mu-\varepsilon}f)^{\frac{1}{2}}\|_{L^{p_{2}(\cdot)r(\cdot)}}&\leq C \|(M^{A}_{\Omega,\mu-\varepsilon}f)\|^{\frac{1}{2}}_{L^{\frac{{p_{2}(\cdot)r(\cdot)}}{2}}}\\ &\leq C \left(\sum_{|\gamma|=m}\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})}\right)^{\frac{1}{2}}\|f\|^{\frac{1}{2}}_{L^{{p_{1}(\cdot)}}(\mathbb{R}^{n})}.\quad\quad\quad \end{align*} Similar way, we concluded that $$ \|(M^{A}_{\Omega,\mu+\varepsilon}f)^{\frac{1}{2}}\|_{L^{p_{2}(\cdot)r'(\cdot)}} \leq C \left(\sum_{|\gamma|=m}\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})}\right)^{\frac{1}{2}}\|f\|^{\frac{1}{2}}_{L^{{p_{1}(\cdot)}}(\mathbb{R}^{n})}, $$ then, we have $$ \|T_{\Omega,\mu}^{A}f(x)\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\leq C \sum_{|\gamma|=m}\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})} \|f \|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}. $$ This completes the proof of Lemma 13.

3. Main results

In this section, we investigate the boundedness of the multilinear fractional integral operator with rough kernel on variable nonhomogeneous Morrey-Herz spaces \({MK}_{q(\cdot),p(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})\).

Theorem 1. Suppose that \({D^{\gamma}A}\in BMO(\mathbb{R}^{n})(|\gamma|=|m|, m\geq1)\). Let \( 0< \mu< n\), \(\Omega\in L^{r}(s^{n-1})\), \(q_{1}(\cdot), q_{2}(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\) with \((q_{2})_{-} \geq (q_{1})_{+}\), and \(p_{1}(\cdot), p_{2}(\cdot) \in \mathfrak{B}(\mathbb{R}^{n})\) satisfy \(0< \mu\leq\frac{n}{( p_{1})_{+}} \), \(\frac{1}{p_{1}(x)} – \frac{1}{p_{2}(x)} = \frac{\mu}{n}\). If \((\lambda_{1})(q_{2})_{+}=(\lambda_{2})(q_{1})_{-}\) and \(\mu- n\delta_{2}+(\lambda_{1})/(q_{1})_{-}< \alpha_{+}< n \delta_{1}+\frac{n}{r}+(\lambda_{1})/(q_{1})_{-} \). Then \(T_{\Omega,\mu}^{A}\) is bounded from \({MK}_{q_{1}(\cdot),p_{1}(\cdot)}^{\alpha_{+}, \lambda_{1}}(\mathbb{R}^{n})\) to \({MK}_{q_{2}(\cdot),p_{2}(\cdot)}^{\alpha(\cdot), \lambda_{2}}(\mathbb{R}^{n})\).

Proof. Let \({D^{\gamma}A}\in BMO(\mathbb{R}^{n})\), \(f \in {{MK}_{q_{1}(\cdot),p_{1}(\cdot)}^{\alpha_{+},\lambda_{1}}(\mathbb{R}^{n})}\), we write \begin{eqnarray*} f(x) = \sum_{j=0}^{\infty}f(x)\chi_{j}(x)= {\sum_{j=0}^{ \infty}} f_{j}(x). \end{eqnarray*} By the definition of \( {{MK}_{q(\cdot),p(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})}\), we have \begin{eqnarray*} \|T^{A}_{\Omega, \mu}(f)\chi_{k}\|_{{MK}_{q_{2}(\cdot),p_{2}(\cdot)}^{\alpha(\cdot),\lambda_{2}}(\mathbb{R}^{n})}= \inf \left\{ \eta> 0: \sup_{k_{0}\in z}{2^{-k_{0}\lambda_{2}}} \sum\limits_{k=0}^{k_{0}}\left\| \left( \frac{2^{k\alpha(\cdot)}| T^{A}_{\Omega , \mu} (f)\chi_{k}|}{\eta}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p_{2}(\cdot)}{q_{2}(\cdot)}}}\leq 1\right\}. \end{eqnarray*} For any \(k_{0}\in z\), we see that \begin{eqnarray*} &&2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}}\left\|\left( \frac{2^{k\alpha(\cdot)}| T^{A}_{\Omega, \mu} (f)\chi_{k}|}{\eta}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p_{2}(\cdot)}{q_{2}(\cdot)}}} \leq 2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}}\left\|\left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=0}^{\infty} T^{A}_{\Omega, \mu} (f_{j})\chi_{k}|}{\eta_{11}+\eta_{12}+\eta_{13}}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p_{2}(\cdot)}{q_{2}(\cdot)}}}\\ &&\leq2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}}\left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=0}^{k-2} T^{A}_{\Omega, \mu} (f_{j})\chi_{k}|}{\eta_{11}}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p_{2}(\cdot)}{q_{2}(\cdot)}}}\quad+2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}}\left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k-1}^{k+1} T^{A}_{\Omega, \mu} (f_{j})\chi_{k}|}{\eta_{12}}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p_{2}(\cdot)}{q_{2}(\cdot)}}}\\ &&+2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}}\left\|\left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k+2}^{\infty} T^{A}_{\Omega, \mu} (f_{j})\chi_{k}|}{\eta_{13}}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p_{2}(\cdot)}{q_{2}(\cdot)}}}, \end{eqnarray*} where \begin{align*} {\eta_{11}} &= \left\|\sum\limits_{j=0}^{k-2}T^{A}_{\Omega, \mu} (f_{j})\chi_{k}\right \|_{{MK}_{q_{2}(\cdot),p_{2}(\cdot)}^{\alpha(\cdot),\lambda_{2}}(\mathbb{R}^{n})}\\ &=\inf \left\{ \eta> 0: \sup_{k_{0}\in z}{2^{-k_{0}\lambda_{2}}} \sum\limits_{k=0}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}| \sum\limits_{j=0}^{k-2} T^{A}_{\Omega , \mu} (f_{j})\chi_{k}|}{\eta}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p_{2}(\cdot)}{q_{2}(\cdot)}}}\leq 1\right\},\\ {\eta_{12}} &= \left\|\sum\limits_{k-1}^{k+1}T^{A}_{\Omega, \mu} (f_{j})\chi_{k}\right \|_{{MK}_{q_{2}(\cdot),p_{2}(\cdot)}^{\alpha(\cdot),\lambda_{2}}(\mathbb{R}^{n})}\\ &=\inf \left\{ \eta> 0 : \sup_{k_{0}\in z}{2^{-k_{0}\lambda_{2}}} \sum\limits_{k=0}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}| \sum\limits_{j=k-1}^{k+1} T^{A}_{\Omega , \mu} (f_{j})\chi_{k}|}{\eta}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p_{2}(\cdot)}{q_{2}(\cdot)}}}\leq 1\right\}, \end{align*} \begin{align*} {\eta_{13}} &= \left\|\sum\limits_{j=k+2}^{\infty}T^{A}_{\Omega, \mu} (f_{j})\chi_{k}\right \|_{{MK}_{q_{2}(\cdot),p_{2}(\cdot)}^{\alpha(\cdot),\lambda_{2}}(\mathbb{R}^{n})}\\ &=\inf \left\{ \eta> 0 : \sup_{k_{0}\in z}{2^{-k_{0}\lambda_{2}}} \sum\limits_{k=0}^{k_{0}} \left\|\left( \frac{2^{k\alpha(\cdot)}| \sum\limits_{j=k+2}^{\infty} T^{A}_{\Omega , \mu} (f_{j})\chi_{k}|}{\eta}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p_{2}(\cdot)}{q_{2}(\cdot)}}}\leq 1\right\}. \end{align*} If \({\eta} = {\eta_{11}} +{\eta_{12}}+{\eta_{13}}\), thus $$ {2^{-k_{0}\lambda_{2}}} \sum\limits_{k=0}^{k_{0}}\left\| \left( \frac{2^{k\alpha(\cdot)}| T^{A}_{\Omega, \mu} (f_{j}) \chi_{k}|}{\eta}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p_{2}(\cdot)}{q_{2}(\cdot)}}} \leq C. $$ That is $$ \|T^{A}_{\Omega , \mu} (f)\chi_{k} \|_{{MK}_{q_{2}(\cdot),p_{2}(\cdot)}^{\alpha(\cdot),\lambda_{2}}(\mathbb{R}^{n})} \leq C \eta \leq C [ \eta_{11} +\eta_{12}+\eta_{13}]. $$ Hence, it suffices to prove $$ \eta_{11} , \eta_{12} ,\eta_{13} \leq C \sum_{|\gamma|=m}\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})} \| f\|_{{MK}_{q_{1}(\cdot),p_{1}(\cdot)}^{\alpha(\cdot),\lambda_{1}}(\mathbb{R}^{n})}, $$ Denote \( \eta_{1} =\| f \|_{{MK}_{q_{1}(\cdot),p_{1}(\cdot)}^{\alpha(\cdot),\lambda_{1}}(\mathbb{R}^{n})}.\)
Now we consider \({\eta_{12}}\) firstly. Applying Lemma 6, noting that \(T_{\Omega,\mu}^{A}\) is bounded on \(L^{p(\cdot)}(\mathbb{R}^{n})\), it follows \begin{align*} &\quad2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}}\left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k-1}^{k+1} T^{A}_{\Omega , \mu} (f_{j}) \chi_{k}|}{\eta_{1} \sum_{|\gamma|=m }\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})}}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p_{2}(\cdot)}{q_{2}(\cdot)}}} \\ &\leq 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}}\left\| \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k-1}^{k+1} T^{A}_{\Omega , \mu} (f_{j}) \chi_{k}|}{\eta_{1} \sum_{|\gamma|=m }\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})}}\right\|^{(q^{1}_{2})_k}_{L^{{p_{2}(\cdot)}}} \\ &\leq 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}}\left(\sum\limits_{j=k-1}^{k+1}\left\| \frac{2^{(k-j)\alpha_{+}}2^{j\alpha_{+}}| T^{A}_{\Omega, \mu} (f_{j}) \chi_{k}|}{\eta_{1} \sum_{|\gamma|=m }\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})}}\right\|_{L^{{p_{2}(\cdot)}}}\right)^{(q^{1}_{2})_k}\\ &\leq 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}}\left(\sum\limits_{j=k-1}^{k+1}\left\| \frac{2^{j\alpha_{+}} |f_{j}| }{\eta_{1} }\right\|_{L^{{p_{1}(\cdot)}}}\right)^{(q^{1}_{2})_k}, \end{align*} where $$ {(q^{1}_{2})_k}= \left\{\begin{array}{ll} (q_{2})_{-},~~~~ ~~~~ \left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k-1}^{k+1} T^{A}_{\Omega, \mu} (f_{j}) \chi_{k}|}{\eta_{1}\sum_{|\gamma|=m }\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})}}\right)^{q_{2}(\cdot)}\right\|_{L^{{p_{2}(\cdot)}}} \leq 1, \\ (q_{2})_{+},~~~~ ~~~~ \left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k-1}^{k+1} T^{A}_{\Omega, \mu} (f_{j}) \chi_{k}|}{\eta_{1}\sum_{|\gamma|=m }\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})}}\right)^{q_{2}(\cdot)}\right\|_{L^{{p_{2}(\cdot)}}} >1. \end{array}\right. $$ Since \(f\in {{MK}_{q_{1}(\cdot),p_{1}(\cdot)}^{\alpha(\cdot),\lambda_{1}}(\mathbb{R}^{n})}\), we have $$ 2^{-k_{0}\lambda_{1}}\left\|\left( \frac{2^{k\alpha_{+}} |f \chi_{k}| }{\eta_{1} }\right)^{q_{1}(\cdot)}\right\|_{L^{\frac{p_{1}(\cdot)}{q_{1}(\cdot)}}}\leq 1. $$ From this and again applying Lemma 6, if \((q_{1})_{+}\leq (q_{2})_{-}\) and \(\lambda_{1}(q_{2})_{+} = \lambda_{2}(q_{1})_{-}\), we can obtain that \begin{align*} &2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}}\left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k-1}^{k+1} T^{A}_{\Omega, \mu} (f_{j}) \chi_{k}|}{\eta_{1} \sum_{|\gamma|=m }\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})}}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p_{2}(\cdot)}{q_{2}(\cdot)}}} \leq C2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}}\left(\left\| \frac{2^{k\alpha_{+}} |f \chi_{k}| }{\eta_{1} }\right\|_{L^{{p_{1}(\cdot)}}}\right)^{(q^{1}_{2})_k}\\ &\leq C2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}}\left\|\left( \frac{2^{k\alpha_{+}} |f \chi_{k}| }{\eta_{1} }\right)^{q_{1}(\cdot)}\right\|^{\frac{(q^{1}_{2})_k}{(q^{1}_{1})_j}}_{L^{\frac{p_{1}(\cdot)}{q_{1}(\cdot)}}} \leq C\sum\limits_{k=0}^{k_{0}}\left\{2^{-k_{0}\lambda_{1}}\left\|\left( \frac{2^{k\alpha_{+}} |f \chi_{k}| }{\eta_{1} }\right)^{q_{1}(\cdot)}\right\|_{L^{\frac{p_{1}(\cdot)}{q_{1}(\cdot)}}} \right\}^{\frac{(q^{1}_{2})_k}{(q^{1}_{1})_j}} \leq C, \end{align*} where $${(q^{1}_{1})_j}= \left\{\begin{array}{ll} (q_{1})_{+},\qquad \left\| \frac{2^{k\alpha_{+}}| f_{ \chi_{k}}|}{\eta_{1}}\right\|_{L^{{p_{1}(\cdot)}}} \leq 1, \\ (q_{1})_{-},\qquad \left\| \frac{2^{k\alpha_{+}}| f_{ \chi_{k}}|}{\eta_{1}}\right\|_{L^{{p_{1}(\cdot)}}} >1. \end{array}\right. $$ This implies that $$ \eta_{12}\leq C \eta_{1}\leq C \sum_{|\gamma|=m}\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})} \| f\|_{\dot{MK}_{q_{1}(\cdot),p_{1}(\cdot)}^{\alpha(\cdot),\lambda_{1}}(\mathbb{R}^{n})}. $$ Now we estimate of \( {\eta_{11}}\). Let \( x \in C_{k} ~ j \leq k-2 \), then \( | x- y|\sim |x|\), we can write that $$ |T^{A}_{\Omega,\mu}f_{j} (x)|\leq C \int_{ C_{j}}\frac{|\Omega( x-y )|}{|x -y|^{n-\mu+m}}|R_{m+1}(A;x,y)||f_{j}(y)|\mathrm{d}y, $$ where $$ A_{k}(x)= A(x) -\sum_{|\gamma|=m}\frac{1}{\gamma!}(D^{\gamma} A)x^{\gamma}, $$ and $$ R_{m+1}(A;x,y) = R_{m+1}(A_{k};x,y),~D^{\gamma}A_{k}(x)= D^{\gamma} A(x)- m_{\gamma_k}(D^{\gamma}A), ~|\gamma|= m. $$ Applying Lemma 10, we see that \begin{align*} |R_{m+1}(A;x,y)|&= |R_{m+1}(A_{k};x,y)|\\ &\leq |R_{m}(A_{k};x,y)| +\sum_{|\gamma|=m}\frac{1}{\gamma!}|D^{\gamma}A_{k}(y)||x-y|^{m}\\ &\leq C |x-y|^{m}\sum_{|\gamma|=m}\left(\left(\frac{1}{|\widetilde{Q}(x,y)|}\int_{\tilde{Q}(x,y)}{|D^{\gamma}A_{k}(z)|^{q}}\mathrm{d}z\right)^{\frac{1}{q}} +\frac{1}{\gamma!}|D^{\gamma} A_{k}(y)|\right) \\ &\leq C |x-y|^{m}\sum_{|\gamma|=m}\left(\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})} + |D^{\gamma} A_{k}(y)| \right). \end{align*} Thus, we get \begin{align*} |T^{A}_{\Omega,\mu}f_{j} (x)|&\leq C \int_{ C_{j}}\frac{|\Omega( x-y )|}{|x -y|^{n-\mu}}\sum_{|\gamma|=m}\left[\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})} + |D^{\gamma} A_{k}(y)| \right]|f_{j}(y)|\mathrm{d}y\\ &\leq C \sum_{|\gamma|=m} \int_{ C_{j}}{\frac{|\Omega( x-y )|}{|x -y|^{n-\mu}}}\left[\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})} + |D^{\gamma} A_{k}(y)| \right]|f_{j}(y)|\mathrm{d}y\\ &\leq C\sum_{|\gamma|=m }\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})}\int_{ C_{j}}{\frac{|\Omega( x-y )|}{|x -y|^{n-\mu}}}|f_{j}(y)|\mathrm{d}y\\ &\,\,\,+ \sum_{|\gamma|=m }\int_{ C_{j}}{\frac{|\Omega( x-y )|}{|x -y|^{n-\mu}}}|D^{\gamma} A_{k}(y)||f_{j}(y)|\mathrm{d}y\\ &=: L_{1} + L_{2}. \end{align*} Applying the generalized Hölder’s inequality, we have $$ L_{1}\leq C \sum_{|\gamma|=m }\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})}\|f_{j}\|_{L^{p_{1}(\cdot)}} \left\|\frac{\Omega( x-y )}{|x -y|^{n-\mu}} \chi_{j}\right\|_{L^{{p’}_{1}(\cdot)}}. $$ If \(\frac{1}{\acute{p}_{1}(\cdot)}= \frac{1}{r} + \frac{1}{\widetilde{p_{1}’}(\cdot)} \), by Lemma 2, then we have \begin{align*} L_{1} &\leq \sum_{|\gamma|=m }\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})} \|f_{j}\|_{L^{p_{1}(\cdot)}}\|\Omega( x-y )\|_{L^{r}}\left\| \frac{\chi_{j}}{|x -y|^{n-\mu}}\right\|_{L^{\widetilde{p’_{1}}(\cdot)}}\end{align*} \begin{align*} &\leq C\sum_{|\gamma|=m }\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})} 2^{-k(n-\mu)}\|f_{j}\|_{L^{p_{1}(\cdot)}} \|\chi_{j}\|_{L^{\widetilde{p_{1}’}(\cdot)}} \left[ \int\limits_{2^{k-2}}^{2^{k}}r^{n-1} \mathrm{d}r\int_{s^{n-1}}|{\Omega(y’)}|^{r}\mathrm{d}{\sigma}(y’)\right]^{\frac{1}{r}}\\ &\leq C\sum_{|\gamma|=m }\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})} 2^{-k(n-\mu)}\|f_{j}\|_{L^{p_{1}(\cdot)}} \|\chi_{j}\|_{L^{\widetilde{p_{1}’}(\cdot)}} 2^{\frac{kn}{r}} \|\Omega\|_{L^{r}(S^{n-1})}\\ &\leq C\sum_{|\gamma|=m }\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})} 2^{-k(n-\mu)}2^{\frac{kn}{r}}\|f_{j}\|_{L^{p_{1}(\cdot)}} \|\chi_{B_j}\|_{L^{\widetilde{p_{1}’}(\cdot)}}. \end{align*} According to Lemma 3 and the formula \( \frac{1}{\widetilde{p_{1}’}(\cdot)}= \frac{1}{{p_{1}’}(\cdot)}-\frac{1}{r}\), we have $$ \|\chi_{B_{j}}\|_{L^{\widetilde{p_{1}’}(\cdot)}} \approx\|\chi_{B_{j}}\|_{L^{p_{1}'(\cdot)}} |B_{j}|^{\frac{-1}{r}} $$ Then, we obtain $$ L_{1} \leq \sum_{|\gamma|=m }\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})}2^{-k(n-\mu)}2^{(k-j)\frac{n}{r}}\|f_{j}\|_{L^{p_{1}(\cdot)}}\|\chi_{B_{j}}\|_{L^{p_{1}'(\cdot)}} . $$ For \(L_{2}\), applying the generalized Hölder’s inequality, we get $$ L_{2}\leq C\sum_{|\gamma|=m }\|f_{j}\|_{L^{p_{1}(\cdot)}} \left\|\frac{\Omega( x-y )}{|x -y|^{n-\mu}} \chi_{j} |D^{\gamma} A_{k}(y)|\right\|_{L^{{p’}_{1}(\cdot)}}. $$ If \(\frac{1}{{p’}_{1}(\cdot)}= \frac{1}{r} + \frac{1}{\widetilde{p_{1}’}(\cdot)} \), by Lemma 3, we have \begin{align*} L_{2}&\leq C\sum_{|\gamma|=m }{2^{-k(n-\mu)}}\|f_{j}\|_{L^{p_{1}(\cdot)}}\|\Omega( x-y )\|_{L^{r}} \left\|\chi_{j}|D^{\gamma} A_{k}(y)| \right\|_{L^{\widetilde{p’_{1}}(\cdot)}}\\ &\leq C{2^{-k(n-\mu)}}\|f_{j}\|_{L^{p_{1}(\cdot)}}\|\Omega( x-y )\|_{L^{r}}\sum_{|\gamma|=m } \left\|\chi_{j}(D^{\gamma} A(x)- m_{\gamma_k}(D^{\gamma}A)) \right\|_{L^{\widetilde{p’_{1}}(\cdot)}}. \end{align*} Similarly, and applying Lemma 7, we conclude that $$ L_{2}\leq \sum_{|\gamma|=m }\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})}(k-j)2^{-k(n-\mu)}2^{(k-j)\frac{n}{r}}\|f_{j}\|_{L^{p_{1}(\cdot)}}\|\chi_{B_{j}}\|_{L^{p_{1}’ (\cdot)}}. $$ Combining the above two estimates about \(L_{1},L_{2}\), we obtain $$ |T^{A}_{\Omega,\mu}f_{j} (x)|\leq C \sum_{|\gamma|=m }\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})}(k-j)2^{-k(n-\mu)}2^{(k-j)\frac{n}{r}}\|f_{j}\|_{L^{p_{1}(\cdot)}}\|\chi_{B_{j}}\|_{L^{p_{1}’ (\cdot)}}. $$ From this and using Lemma 6, we deduce that \begin{align*} 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}}\left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=0}^{k-2} T^{A}_{\Omega , \mu} (f_{j}) \chi_{k}|}{\eta_{1} \sum_{|\gamma|=m }\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})}}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p_{2}(\cdot)}{q_{2}(\cdot)}}} \leq 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}}\left\| \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=0}^{k-2} T^{A}_{\Omega , \mu} (f_{j}) \chi_{k}|}{\eta_{1} \sum_{|\gamma|=m }\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})}}\right\|^{(q_{2}^{2})_k}_{L^{{p_{2}(\cdot)}}}, \end{align*} where $$ {(q^{2}_{2})_k}= \left\{\begin{array}{ll} (q_{2})_{-},\qquad \left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=0}^{k-2} T^{A}_{\Omega , \mu} (f_{j}) \chi_{k}|}{\eta_{1}\sum_{|\gamma|=m }\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})}}\right)^{q_{2}(\cdot)}\right\|_{L^{{p_{2}(\cdot)}}} \leq 1, \\ (q_{2})_{+},\qquad \left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=0}^{k-2} T^{A}_{\Omega , \mu} (f_{j}) \chi_{k}|}{\eta_{1}\sum_{|\gamma|=m }\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})}}\right)^{q_{2}(\cdot)}\right\|_{L^{{p_{2}(\cdot)}}} >1. \end{array}\right. $$ Noting that if \(\frac{1}{p_{1}(\cdot)}-\frac{1}{p_{2}(\cdot)}=\frac{\mu}{n}\), then \(C_{1}|B|^{\frac{\mu}{n}}\|\chi_{B}\|_{L^{p_{2}(\cdot)}}\leq\|\chi_{B}\|_{L^{p_{1}(\cdot)}}\leq C_{2}|B|^{\frac{\mu}{n}} \|\chi_{B}\|_{L^{p_{2}(\cdot)}}\) (see[22], p.370).
Therefore, together with this and applying Lemma 4 and Lemma 5, we have \begin{eqnarray*} &&2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}}\left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=0}^{k-2} T^{A}_{\Omega , \mu} (f_{j}) \chi_{k}|}{\eta_{1} \sum_{|\gamma|=m }\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})}}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p_{2}(\cdot)}{q_{2}(\cdot)}}}\\ &&\leq 2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}} \left[ 2^{k\alpha(\cdot)}\sum\limits_{j=0}^{k-2}2^{-k(n-\mu)}2^{(k-j)\frac{n}{r}}(k-j) \left\|\frac{|f_{j}|}{\eta_{1}}\right\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{j}}\|_{L^{p_{1}'(\cdot)}} \|\chi_{B_{k}}\|_{L^{p_{2}(\cdot)}} \right]^{(q^{2}_{2})_k}\\ &&\leq C 2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}} \left[ 2^{k\alpha(\cdot)}\sum\limits_{j=0}^{k-2}2^{-k(n-\mu)}2^{(k-j)\frac{n}{r}}(k-j) \left\|\frac{|f_{j}|}{\eta_{1}}\right\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{j}}\|_{L^{p_{1}'(\cdot)}} |B_{k}|^{-\frac{\mu}{n}}\|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}} \right]^{(q^{2}_{2})_k}\\ &&\leq C 2^{-k_{0}\lambda_{2}} \sum\limits_{k=-\infty}^{\infty} \left[ 2^{k\alpha(\cdot)}\sum\limits_{j=0}^{k-2}(k-j)2^{(k-j)\frac{n}{r}} \left\|\frac{|f_{j}|}{\eta_{1}}\right\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \frac{\|\chi_{B_j}\|_{L^{p_{1}'(\cdot)}}}{\|\chi_{B_k}\|_{L^{p_{1}'(\cdot)}}} \right]^{(q^{2}_{2})_k}\\ &&\leq C 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{\infty} \left[ 2^{k\alpha(\cdot)}\sum\limits_{j=0}^{k-2}(k-j)2^{(j-k)(n\delta_{1}-\frac{n}{r}}) \left\|\frac{|f_{j}|}{\eta_{1}}\right\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \right]^{(q^{2}_{2})_k}\\ &&\leq C 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{\infty} \left[ \sum\limits_{j=0}^{k-2}(k-j) 2^{(j-k)(n\delta_{1}-\frac{n}{r}-\alpha_{+}}) \left\|\frac{ 2^{j\alpha_{+}}|f\chi_{k}|}{\eta_{1}}\right\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \right]^{(q^{2}_{2})_k}. \end{eqnarray*} Since \(f \in {{MK}_{q_{1}(\cdot),p_{1}(\cdot)}^{\alpha_{+},\lambda_{1}}(\mathbb{R}^{n})}\), \((\lambda_{2})/(q_{2})_{+}=(\lambda_{1})/(q_{1})_{-}\) and \(\alpha_{+}< n\delta_{1}+\frac{n}{r}+(\lambda_{1})/(q_{1})_{-}\), we have \begin{eqnarray*} &&2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}}\left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=0}^{k-2} T^{A}_{\Omega , \mu} (f_{j}) \chi_{k}|}{\eta_{1} \sum_{|\gamma|=m }\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})}}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p_{2}(\cdot)}{q_{2}(\cdot)}}}\\ &&\leq C 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{\infty} \left[ \sum\limits_{j=0}^{k-2}(k-j) 2^{(j-k)(n\delta_{1}-\frac{n}{r}-\alpha_{+}}) \left\|\left(\frac{ 2^{j\alpha_{+}}|f\chi_{k}|}{\eta_{1}}\right)^{q_{1}(\cdot)}\right\|^{\frac{1}{(q^{2}_{1})j}}_{L^\frac{p_{1}(\cdot)}{q_{1}(\cdot)}(\mathbb{R}^{n})} \right]^{(q^{2}_{2})_k}\\ &&\leq C 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{\infty} \left[ \sum\limits_{j=0}^{k-2}(k-j) 2^{(j-k)(n\delta_{1}-\frac{n}{r}-\alpha_{+}}) \left( 2^{j\lambda}2^{-j\lambda}\sum\limits^{j}_{n=0} \left\|\left(\frac{ 2^{n\alpha_{+}}|f\chi_{n}|}{\eta_{1}}\right)^{q_{1}(\cdot)}\right\|_{L^\frac{p_{1}(\cdot)}{q_{1}(\cdot)}(\mathbb{R}^{n})} \right)^{\frac{1}{(q^{2}_{1})j}} \right]^{(q^{2}_{2})_k}\\ &&\leq C \sum\limits_{k=0}^{\infty}2^{(k-k_{0})\lambda_{2}} \\ &&\times\left[ \sum\limits_{j=0}^{k-2}(k-j) 2^{(j-k)((\lambda_{1})/(q_{1})_{-} +n\delta_{1}-\frac{n}{r}-\alpha_{+})}\left( 2^{-j\lambda}\sum\limits^{j}_{n=0} \left\|\left(\frac{ 2^{n\alpha_{+}}|f\chi_{n}|}{\eta_{1}}\right)^{q_{1}(\cdot)}\right\|_{L^\frac{p_{1}(\cdot)}{q_{1}(\cdot)}(\mathbb{R}^{n})} \right)^{\frac{1}{(q^{2}_{1})j}} \right]^{(q^{2}_{2})_k}\\ &&\leq C \sum\limits_{k=0}^{\infty}2^{(k-k_{0})\lambda_{2}} \left[ \sum\limits_{j=0}^{k-2}(k-j) 2^{(j-k)((\lambda_{1})/(q_{1})_{-} +n\delta_{1}-\frac{n}{r}-\alpha_{+})} \right]^{(q^{2}_{2})_k} \leq C, \end{eqnarray*} where $$ {(q^{2}_{1})_j}= \left\{\begin{array}{ll} (q_{1})_{-},\qquad \left\| \frac{2^{j\alpha_{+}}| f_{ \chi_{j}}|}{\eta_{1}}\right\|_{L^{{p_{1}(\cdot)}}} \leq 1, \\ (q_{1})_{+},\qquad \left\| \frac{2^{j\alpha_{+}}| f_{ \chi_{j}}|}{\eta_{1}}\right\|_{L^{{p_{1}(\cdot)}}} >1. \end{array}\right. $$ This implies that $$ \eta_{11}\leq C \eta_{1}\sum_{|\gamma|=m}\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})} = \sum_{|\gamma|=m}\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})} \| f\|_{\dot{MK}_{q_{1}(\cdot),p_{1}(\cdot)}^{\alpha(\cdot),\lambda_{1}}(\mathbb{R}^{n})}. $$ Finally, we estimate of \(\eta_{13}\). Let \( x \in C_{k} , ~ j \geq k+2 \), then \( | x- y|\sim |y|\), by an argument similar to used in \(\eta_{11}\), we have $$ |T^{A}_{\Omega,\mu}f_{j} (x)|\leq C \sum_{|\gamma|=m }\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})}(k-j)2^{-j(n-\mu)}\|f_{j}\|_{L^{p_{1}(\cdot)}}\|\chi_{B_{j}}\|_{L^{p_{1}’ (\cdot)}}. $$ Applying Lemma 6, we have \begin{align*} 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}}\left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{k+2}^{\infty} T^{A}_{\Omega , \mu} (f_{j}) \chi_{k}|}{\eta_{1} \sum_{|\gamma|=m }\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})}}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p_{2}(\cdot)}{q_{2}(\cdot)}}} \leq 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}}\left\| \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k+2}^{\infty} T^{A}_{\Omega , \mu} (f_{j}) \chi_{k}|}{\eta_{1} \sum_{|\gamma|=m }\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})}}\right\|^{(q_{2}^{3})_k}_{L^{{p_{2}(\cdot)}}}, \end{align*} where $$ {(q^{3}_{2})_k}= \left\{\begin{array}{ll} (q_{2})_{-},\qquad \left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k+2}^{\infty} T^{A}_{\Omega , \mu} (f_{j}) \chi_{k}|}{\eta_{1}\sum_{|\gamma|=m }\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})}}\right)^{q_{2}(\cdot)}\right\|_{L^{{p_{2}(\cdot)}}} \leq 1, \\ (q_{2})_{+},\qquad \left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k+2}^{\infty} T^{A}_{\Omega , \mu} (f_{j}) \chi_{k}|}{\eta_{1}\sum_{|\gamma|=m }\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})}}\right)^{q_{2}(\cdot)}\right\|_{L^{{p_{2}(\cdot)}}} >1. \end{array}\right. $$ Therefore, applying Lemma 4 and Lemma 5, we get \begin{align*} &2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}} \left[ 2^{k\alpha(\cdot)}\sum\limits_{j=k+2}^{\infty}2^{-j(n-\mu)}(k-j) \left\|\frac{|f_{j}|}{\eta_{1}}\right\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{j}}\|_{L^{p_{1}'(\cdot)}} \|\chi_{B_{k}}\|_{L^{p_{2}(\cdot)}} \right]^{(q^{3}_{2})_k}\\ &\leq C 2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}} \left[ 2^{k\alpha(\cdot)}\sum\limits_{j=k+2}^{\infty}2^{-j(n-\mu)}(k-j) \left\|\frac{|f_{j}|}{\eta_{1}}\right\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{j}}\|_{L^{p_{1}'(\cdot)}} 2^{-\mu k}\|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}} \right]^{(q^{3}_{2})_k}\\ &\leq C 2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}} \left[ 2^{k\alpha(\cdot)}\sum\limits_{j=k+2}^{\infty}2^{(j-k)\mu}(k-j) \left\|\frac{|f_{j}|}{\eta_{1}}\right\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} 2^{-jn}\|\chi_{B_{j}}\|_{L^{p_{1}'(\cdot)}} \|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}} \right]^{(q^{3}_{2})_k}\\ &\leq C 2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}} \left[ 2^{k\alpha(\cdot)}\sum\limits_{j=k+2}^{\infty}2^{(j-k)\mu}(k-j) \left\|\frac{|f_{j}|}{\eta_{1}}\right\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \frac{\|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}}}{\|\chi_{B_{j}}\|_{L^{p_{1}(\cdot)}}} \right]^{(q^{3}_{2})_k}\\ &\leq C 2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}} \left[ 2^{k\alpha(\cdot)}\sum\limits_{j=k+2}^{\infty}(k-j) 2^{(k-j)(n\delta_{2}-\mu)}\left\|\frac{|f_{j}|}{\eta_{1}}\right\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \right]^{(q^{3}_{2})_k}. \end{align*} Notice that \(f \in {{MK}_{q_{1}(\cdot),p_{1}(\cdot)}^{\alpha_{+},\lambda_{1}}(\mathbb{R}^{n})}\), \((\lambda_{2})/(q_{2})_{+}=(\lambda_{1})/(q_{1})_{-}\) and \(\alpha_{+}>\mu- n\delta_{2}+(\lambda_{1})/(q_{1})_{-}\), we have \begin{align*} &2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}}\left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k+2}^{\infty} T^{A}_{\Omega , \mu} (f_{j}) \chi_{k}|}{\eta_{1} \sum_{|\gamma|=m }\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})}}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p_{2}(\cdot)}{q_{2}(\cdot)}}}\\ &\leq C 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{\infty} \left[ \sum\limits_{j=k+2}^{\infty}(k-j)2^{(k-j)(n\delta_{2}-\mu+\alpha_{+})} \left\|\left(\frac{ 2^{j\alpha_{+}}|f\chi_{k}|}{\eta_{1}}\right)^{q_{1}(\cdot)}\right\|^{\frac{1}{(q^{3}_{1})j}}_{L^\frac{p_{1}(\cdot)} {q_{1}(\cdot)}(\mathbb{R}^{n})} \right]^{(q^{3}_{2})_k} \end{align*} \begin{align*} &\leq C 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{\infty} \left[ \sum\limits_{j=k+2}^{\infty}(k-j) 2^{(k-j)(n\delta_{2}-\mu+\alpha_{+}}) \left( 2^{j\lambda}2^{-j\lambda}\sum\limits^{j}_{n=0} \left\|\left(\frac{ 2^{n\alpha_{+}}|f\chi_{n}|}{\eta_{1}}\right)^{q_{1}(\cdot)}\right\|_{L^\frac{p_{1}(\cdot)}{q_{1}(\cdot)}(\mathbb{R}^{n})} \right)^{\frac{1}{(q^{3}_{1})j}} \right]^{(q^{3}_{2})_k}\\ &\leq C \sum\limits_{k=0}^{\infty}2^{(k-k_{0})\lambda_{2}} \\&\qquad\times\left[ \sum\limits_{j=k+2}^{\infty}(k-j) 2^{(k-j)( n\delta_{2}-\mu -(\lambda_{1})/(q_{1})_{-} +\alpha_{+})} \left( 2^{-j\lambda}\sum\limits^{j}_{n=0} \left\|\left(\frac{ 2^{n\alpha_{+}}|f\chi_{n}|}{\eta_{1}}\right)^{q_{1}(\cdot)}\right\|_{L^\frac{p_{1}(\cdot)}{q_{1}(\cdot)}(\mathbb{R}^{n})} \right)^{\frac{1}{(q^{3}_{1})j}} \right]^{(q^{3}_{2})_k}\\ &\leq C \sum\limits_{k=0}^{\infty}2^{(k-k_{0})\lambda_{2}} \left[ \sum\limits_{j=k+2}^{\infty}(k-j) 2^{(k-j)( n\delta_{2}-\mu -(\lambda_{1})/(q_{1})_{-} -\alpha_{+})} \right]^{(q^{3}_{2})_k} \leq C, \end{align*} where $$ {(q^{3}_{1})_j}= \left\{\begin{array}{ll} (q_{1})_{-},\qquad \left\| \frac{2^{j\alpha_{+}}| f_{ \chi_{j}}|}{\eta_{1}}\right\|_{L^{{p_{1}(\cdot)}}} \leq 1, \\ (q_{1})_{+},\qquad \left\| \frac{2^{j\alpha_{+}}| f_{ \chi_{j}}|}{\eta_{1}}\right\|_{L^{{p_{1}(\cdot)}}} >1. \end{array}\right. $$ This implies that $$ \eta_{13}\leq C \eta_{1}\sum_{|\gamma|=m}\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})} = \sum_{|\gamma|=m}\|D^{\gamma}A\|_{BMO(\mathbb{R}^{n})} \| f\|_{{MK}_{q_{1}(\cdot),p_{1}(\cdot)}^{\alpha(\cdot),\lambda_{1}}(\mathbb{R}^{n})}. $$ The proof of Theorem 1 is finished.

Theorem 2. Suppose that \({D^{\gamma}A}\in \dot{\wedge}_{\beta}(|\gamma|=|m|, m\geq1)\). Let \( 0< \mu< n,\Omega\in L^{r}(s^{n-1})\), \(q_{1}(\cdot),~q_{2}(\cdot) \in \mathcal{P}(\mathbb{R}^{n})\) with \( (q_{2})_{-} \geq (q_{1})_{+}\), and \(p_{1}(\cdot), p_{2}(\cdot)\in \mathfrak{B}(\mathbb{R}^{n})\) satisfy \(0< \mu+\beta\leq\frac{n}{( p_{1})_{+}}\), \(\frac{1}{p_{1}(x)} – \frac{1}{p_{2}(x)} = \frac{\mu+\beta}{n}\). If \((\lambda_{1})(q_{2})_{+}=(\lambda_{2})(q_{1})_{-}\) and \((\mu+\beta)- n\delta_{2}+(\lambda_{1})/(q_{1})_{-}< \alpha_{+}< n\delta_{1}+\frac{n}{r}+(\lambda_{1})/(q_{1})_{-} \). Then \(T_{\Omega,\mu}^{A}\) is bounded from \({MK}_{q_{1}(\cdot),p_{1}(\cdot)}^{\alpha_{+}, \lambda_{1}}(\mathbb{R}^{n})\) to \({MK}_{q_{2}(\cdot),p_{2}(\cdot)}^{\alpha(\cdot), \lambda_{2}}(\mathbb{R}^{n})\).

Proof. Let \({D^{\gamma}A}\in \dot{\wedge}_{\beta}(\mathbb{R}^{n})\), \(f \in {{MK}_{q_{1}(\cdot),p_{1}(\cdot)}^{\alpha_{+},\lambda_{1}}(\mathbb{R}^{n})}\), we write $$ f(x) = \sum_{j=0}^{\infty}f(x)\chi_{j}= {\sum_{j=0}^{ \infty}} f_{j}(x). $$ By the definition of \( {{MK}_{q(\cdot),p(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})}\), we have $$ \left\|T^{A}_{\Omega, \mu} (f)\chi_{k}\right \|_{{MK}_{q_{2}(\cdot),p_{2}(\cdot)}^{\alpha(\cdot),\lambda_{2}}(\mathbb{R}^{n})}= inf \left\{ \eta> 0 : \sup_{k_{0}\in z}{2^{-k_{0}\lambda_{2}}} \sum\limits_{k=0}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}| T^{A}_{\Omega , \mu} (f)\chi_{k}|}{\eta}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p_{2}(\cdot)}{q_{2}(\cdot)}}}\leq 1\right\}. $$ For any \(k_{0}\in z\), we see that \begin{eqnarray*} &2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}}\left\| \left( \frac{2^{k\alpha(\cdot)}| T^{A}_{\Omega, \mu} (f)\chi_{k}|}{\eta}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p_{2}(\cdot)}{q_{2}(\cdot)}}} \leq 2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}}\left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=0}^{\infty} T^{A}_{\Omega, \mu} (f_{j})\chi_{k}|}{\eta_{21}+\eta_{22}+\eta_{23}}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p_{2}(\cdot)}{q_{2}(\cdot)}}}\\ &\leq 2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}}\left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=0}^{k-2} T^{A}_{\Omega, \mu} (f_{j})\chi_{k}|}{\eta_{21}}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p_{2}(\cdot)}{q_{2}(\cdot)}}}+2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}}\left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k-1}^{k+1} T^{A}_{\Omega, \mu} (f_{j})\chi_{k}|}{\eta_{22}}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p_{2}(\cdot)}{q_{2}(\cdot)}}}\\ & +2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}}\left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k+2}^{\infty} T^{A}_{\Omega, \mu} (f_{j})\chi_{k}|}{\eta_{23}}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p_{2}(\cdot)}{q_{2}(\cdot)}}}, \end{eqnarray*} where \begin{align*} {\eta_{21}} &= \left\|\sum\limits_{j=0}^{k-2}T^{A}_{\Omega, \mu} (f_{j})\chi_{k}\right \|_{{MK}_{q_{2}(\cdot),p_{2}(\cdot)}^{\alpha(\cdot),\lambda_{2}}(\mathbb{R}^{n})} \\ &=\inf \left\{ \eta> 0 : \sup_{k_{0}\in z}{2^{-k_{0}\lambda_{2}}} \sum\limits_{k=0}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}| \sum\limits_{j=0}^{k-2} T^{A}_{\Omega , \mu} (f_{j})\chi_{k}|}{\eta}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p_{2}(\cdot)}{q_{2}(\cdot)}}}\leq 1\right\}, \end{align*} \begin{align*} {\eta_{22}} &= \left\|\sum\limits_{k-1}^{k+1}T^{A}_{\Omega, \mu} (f_{j})\chi_{k}\right \|_{{MK}_{q_{2}(\cdot),p_{2}(\cdot)}^{\alpha(\cdot),\lambda_{2}}(\mathbb{R}^{n})}\\ &=\inf \left\{ \eta> 0 : \sup_{k_{0}\in z}{2^{-k_{0}\lambda_{2}}} \sum\limits_{k=0}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}| \sum\limits_{j=k-1}^{k+1} T^{A}_{\Omega , \mu} (f_{j})\chi_{k}|}{\eta}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p_{2}(\cdot)}{q_{2}(\cdot)}}}\leq 1\right\}, \end{align*} \begin{align*} {\eta_{23}} &= \left\|\sum\limits_{j=k+2}^{\infty}T^{A}_{\Omega, \mu} (f_{j})\chi_{k}\right \|_{{MK}_{q_{2}(\cdot),p_{2}(\cdot)}^{\alpha(\cdot),\lambda_{2}}(\mathbb{R}^{n})}\\ &=\inf \left\{ \eta> 0: \sup_{k_{0}\in z}{2^{-k_{0}\lambda_{2}}} \sum\limits_{k=0}^{k_{0}} \left\| \left( \frac{2^{k\alpha(\cdot)}| \sum\limits_{j=k+2}^{\infty} T^{A}_{\Omega , \mu} (f_{j})\chi_{k}|}{\eta}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p_{2}(\cdot)}{q_{2}(\cdot)}}}\leq 1\right\}. \end{align*} And \({\eta} = {\eta_{21}} +{\eta_{22}}+{\eta_{23}}\), thus $$ {2^{-k_{0}\lambda_{2}}} \sum\limits_{k=0}^{k_{0}}\left\| \left( \frac{2^{k\alpha(\cdot)}| T^{A}_{\Omega, \mu} (f_{j}) \chi_{k}|}{\eta}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p_{2}(\cdot)}{q_{2}(\cdot)}}} \leq C. $$ So, we have $$ \|T^{A}_{\Omega , \mu} (f)\chi_{k} \|_{{MK}_{q_{2}(\cdot),p_{2}(\cdot)}^{\alpha(\cdot),\lambda_{2}}(\mathbb{R}^{n})} \leq C \eta \leq C [ \eta_{21} +\eta_{22}+\eta_{23}]. $$ Hence $$ \eta_{21}, \eta_{22},\eta_{23} \leq C \sum\limits_{|\gamma|=m}\|D^{\gamma}A\|_{\dot{\wedge}_{\beta}} \|f\|_{{MK}_{q_{1}(\cdot),p_{1}(\cdot)}^{\alpha(\cdot),\lambda_{1}}(\mathbb{R}^{n})}. $$ Denote \( \eta_{1} =\| f \|_{{MK}_{q_{1}(\cdot),p_{1}(\cdot)}^{\alpha(\cdot),\lambda_{1}}(\mathbb{R}^{n})} \).
Now we consider \({\eta_{22}}\) firstly. Noting that \(T_{\Omega,\mu}^{A}\) is bounded on \(L^{p(\cdot)}\) ( Theorem 5 in [14]), as argued about \(\eta_{12}\) in proof of Theorem 1, we can get $$ 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}}\left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k-1}^{k+1} T^{A}_{\Omega , \mu} (f_{j}) \chi_{k}|}{\eta_{1} \sum\limits_{|\gamma|=m}\|D^{\gamma}A\|_{\dot{\wedge}_{\beta}}}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p_{2}(\cdot)} {q_{2}(\cdot)}}} \leq C. $$ This implies that $$ \eta_{22}\leq C \eta_{1}\leq C \sum\limits_{|\gamma|=m}\|D^{\gamma}A\|_{\dot{\wedge}_{\beta}} \| f\|_{\dot{MK}_{q_{1}(\cdot),p_{1}(\cdot)}^{\alpha(\cdot),\lambda_{1}}(\mathbb{R}^{n})}. $$ Next, we consider \({\eta_{21}}\). Let \( x \in C_{k}\), \(j \leq k-2 \), then \( | x- y|\sim |x|\), we get $$ |T^{A}_{\Omega,\mu}f_{j} (x)|\leq \int_{ C_{j}}\frac{|\Omega( x-y )f_{j}(y)|}{|x -y|^{n-\mu+m}}|R_{m+1}(A;x,y)|\mathrm{d}y. $$ By Lemma 9, and applying Hölder’s inequality, we have \begin{align*} |T^{A}_{\Omega,\mu}f_{j} (x)|&\leq C \sum\limits_{|\gamma|=m}\|D^{\gamma}A\|_{\dot{\wedge}_{\beta}} \int_{ C_{j}}\frac{|\Omega( x-y )f_{j}(y)|}{|x -y|^{n-(\mu+\beta)}}\mathrm{d}y\\ &\leq C \sum\limits_{|\gamma|=m}\|D^{\gamma}A\|_{\dot{\wedge}_{\beta}}\|f_{j}\|_{L^{p_{1}(\cdot)}} \left\|\frac{\Omega( x-y )}{|x -y|^{n-(\mu+\beta)}} \chi_{j}\right\|_{L^{{p’}_{1}(\cdot)}}. \end{align*} In the same way as we estimated \(L_{1}\) in Theorem 1, we obtain that $$ |T^{A}_{\Omega,\mu}f_{j} (x)|\leq C \sum\limits_{|\gamma|=m}\|D^{\gamma}A\|_{\dot{\wedge}_{\beta}} 2^{-k(n-(\mu+\beta))} 2^{(k-j)\frac{n}{r}}\|f_{j}\|_{L^{p_{1}(\cdot)}}\|\chi_{B_{j}}\|_{L^{p_{1}'(\cdot)}}. $$ Applying Lemma 6, we get that $$ 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}}\left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=0}^{k-2} T^{A}_{\Omega , \mu} (f_{j}) \chi_{k}|}{\eta_{1} \sum\limits_{|\gamma|=m}\|D^{\gamma}A\|_{\dot{\wedge}_{\beta}}}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p_{2}(\cdot)} {q_{2}(\cdot)}}} \leq 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}}\left\| \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=0}^{k-2} T^{A}_{\Omega , \mu} (f_{j}) \chi_{k}|}{\eta_{1} \sum\limits_{|\gamma|=m}\|D^{\gamma}A\|_{\dot{\wedge}_{\beta}}}\right\|^{(q_{2}^{1})_k}_{L^{{p_{2}(\cdot)}}}. $$ Where $$ {(q^{1}_{2})_k}= \left\{\begin{array}{ll} (q_{2})_{-},\qquad \left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=0}^{k-2} T^{A}_{\Omega , \mu} (f_{j}) \chi_{k}|}{\eta_{1} \sum\limits_{|\gamma|=m}\|D^{\gamma}A\|_{\dot{\wedge}_{\beta}}}\right)^{q_{2}(\cdot)}\right\|_{L^{{p_{2}(\cdot)}}} \leq 1, \\ (q_{2})_{+},\qquad \left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=0}^{k-2} T^{A}_{\Omega , \mu} (f_{j}) \chi_{k}|}{\eta_{1} \sum\limits_{|\gamma|=m}\|D^{\gamma}A\|_{\dot{\wedge}_{\beta}}}\right)^{q_{2}(\cdot)}\right\|_{L^{{p_{2}(\cdot)}}} >1. \end{array}\right. $$ Since \(\frac{1}{p_{1}(\cdot)}-\frac{1}{p_{2}(\cdot)}=\frac{\mu+\beta}{n}\), then \(\|\chi_{B}\|_{L^{p_{2}(\cdot)}}\leq C 2^{-k(\mu+\beta)} \|\chi_{B}\|_{L^{p_{1}(\cdot)}}\) (see[22], P. 370). Therefore, applying Lemma 4 and Lemma 5, we deduce that \begin{align*} &\quad2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}}\left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=0}^{k-2} T^{A}_{\Omega , \mu} (f_{j}) \chi_{k}|}{\eta_{1} \sum\limits_{|\gamma|=m}\|D^{\gamma}A\|_{\dot{\wedge}_{\beta}}}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p_{2}(\cdot)} {q_{2}(\cdot)}}}\\ &\leq C 2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}} \left[ 2^{k\alpha(\cdot)}\sum\limits_{j=0}^{k-2}2^{-k(n-(\mu+\beta))}2^{(k-j)\frac{n}{r}} \left\|\frac{|f_{j}|}{\eta_{1}}\right\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{j}}\|_{L^{p_{1}'(\cdot)}} \|\chi_{B_{k}}\|_{L^{p_{2}(\cdot)}} \right]^{(q^{1}_{2})_k}\\ &\leq C 2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}} \left[ 2^{k\alpha(\cdot)}\sum\limits_{j=0}^{k-2}2^{-k(n-(\mu+\beta))}2^{(k-j)\frac{n}{r}} \left\|\frac{|f_{j}|}{\eta_{1}}\right\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{j}}\|_{L^{p_{1}'(\cdot)}} 2^{-k(\mu+\beta)}\|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}} \right]^{(q^{1}_{2})_k} \\ &\leq C 2^{-k_{0}\lambda_{2}} \sum\limits_{k=-\infty}^{\infty} \left[ 2^{k\alpha(\cdot)}\sum\limits_{j=0}^{k-2}2^{(k-j)\frac{n}{r}} \left\|\frac{|f_{j}|}{\eta_{1}}\right\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \frac{\|\chi_{B_j}\|_{L^{p_{1}'(\cdot)}}}{\|\chi_{B_k}\|_{L^{p_{1}'(\cdot)}}} \right]^{(q^{1}_{2})_k}\\ &\leq C 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{\infty} \left[ 2^{k\alpha(\cdot)}\sum\limits_{j=0}^{k-2}2^{(j-k)(n\delta_{1}-\frac{n}{r}}) \left\|\frac{|f_{j}|}{\eta_{1}}\right\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \right]^{(q^{1}_{2})_k}\\ &\leq C 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{\infty} \left[ \sum\limits_{j=0}^{k-2} 2^{(j-k)(n\delta_{1}-\frac{n}{r}-\alpha_{+}}) \left\|\frac{ 2^{j\alpha_{+}}|f\chi_{k}|}{\eta_{1}}\right\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \right]^{(q^{1}_{2})_k} \end{align*} \begin{align*} \leq C 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{\infty} \left[ \sum\limits_{j=0}^{k-2} 2^{(j-k)(n\delta_{1}-\frac{n}{r}-\alpha_{+}}) \left\|\left(\frac{ 2^{j\alpha_{+}}|f\chi_{k}|}{\eta_{1}}\right)^{q_{1}(\cdot)}\right\|^{\frac{1}{(q^{1}_{1})j}}_{L^\frac{p_{1}(\cdot)}{q_{1}(\cdot)}(\mathbb{R}^{n})} \right]^{(q^{1}_{2})_k}. \end{align*} Notice that \(f \in {{MK}_{q_{1}(\cdot),p_{1}(\cdot)}^{\alpha_{+},\lambda_{1}}(\mathbb{R}^{n})}\), \((\lambda_{2})/(q_{2})_{+}=(\lambda_{1})/(q_{1})_{-}\) and \(\alpha_{+}< n\delta_{1}+\frac{n}{r}+(\lambda_{1})/(q_{1})_{-}\). In the same way as we estimated \( \eta_{11}\) in Theorem 1, we obtain that \begin{align*} &2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}}\left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=0}^{k-2} T^{A}_{\Omega , \mu} (f_{j}) \chi_{k}|}{\eta_{1} \sum\limits_{|\gamma|=m}\|D^{\gamma}A\|_{\dot{\wedge}_{\beta}}}\right)^{q_{2}(\cdot)}\right\| _{L^{\frac{p_{2}(\cdot)}{q_{2}(\cdot)}}}\\ &\leq C \sum\limits_{k=0}^{\infty}2^{(k-k_{0})\lambda_{2}} \left[ \sum\limits_{j=0}^{k-2} 2^{(j-k)((\lambda_{1})/(q_{1})_{-} +n\delta_{1}-\frac{n}{r}-\alpha_{+})}\right. \left. \times \left( 2^{-j\lambda}\sum\limits^{j}_{n=0} \left\|\left(\frac{ 2^{n\alpha_{+}}|f\chi_{n}|}{\eta_{1}}\right)^{q_{1}(\cdot)}\right\|_{L^\frac{p_{1}(\cdot)}{q_{1}(\cdot)}(\mathbb{R}^{n})} \right)^{\frac{1}{(q^{1}_{1})j}} \right]^{(q^{1}_{2})_k}\\ &\leq C, \end{align*} where $$ {(q^{1}_{1})_j}= \left\{\begin{array}{ll} (q_{1})_{-},\qquad \left\| \frac{2^{j\alpha_{+}}| f_{ \chi_{j}}|}{\eta_{1}}\right\|_{L^{{p_{1}(\cdot)}}} \leq 1, \\ (q_{1})_{+},\qquad \left\| \frac{2^{j\alpha_{+}}| f_{ \chi_{j}}|}{\eta_{1}}\right\|_{L^{{p_{1}(\cdot)}}} >1. \end{array}\right. $$ This implies that $$ \eta_{21}\leq C \eta_{1}\sum\limits_{|\gamma|=m}\|D^{\gamma}A\|_{\dot{\wedge}_{\beta}} = \sum\limits_{|\gamma|=m}\|D^{\gamma}A\|_{\dot{\wedge}_{\beta}} \| f\|_{\dot{MK}_{q_{1}(\cdot),p_{1}(\cdot)}^{\alpha(\cdot),\lambda_{1}}(\mathbb{R}^{n})}. $$ Finally, we consider \(\eta_{23}\). Let \( x \in C_{k},~ j \geq k+2 \), then \( | x- y|\sim |y|\), we have $$ |T^{A}_{\Omega,\mu}f_{j} (x)|\leq \int_{ C_{j}}\frac{|\Omega( x-y )||f_{j}(y)|}{|x -y|^{n-\mu+m}}|R_{m+1}(A;x,y)|\mathrm{d}y. $$ Applying Lemma 10 and Hölder’s inequality, then we have \begin{align*} |T^{A}_{\Omega,\mu}f_{j} (x)|&\leq C \sum\limits_{|\gamma|=m}\|D^{\gamma}A\|_{\dot{\wedge}_{\beta}} \int_{ C_{j}}\frac{|\Omega( x-y )||f_{j}(y)|}{|x -y|^{n-(\mu+\beta)}}\mathrm{d}y\\ &\leq C \sum\limits_{|\gamma|=m}\|D^{\gamma}A\|_{\dot{\wedge}_{\beta}} {2^{-j(n-(\mu+\beta))}}\|f_{j}\|_{L^{p_{1}(\cdot)}} \left\|\Omega( x-y ) \chi_{j}\right\|_{L^{{p’}_{1}(\cdot)}}\\ &\leq C \sum\limits_{|\gamma|=m}\|D^{\gamma}A\|_{\dot{\wedge}_{\beta}} {2^{-j(n-(\mu+\beta))}}\|f_{j}\|_{L^{p_{1}(\cdot)}}\|\chi_{B_{j}}\|_{L^{p_{1}'(\cdot)}}. \end{align*} From this and applying Lemma 4 and Lemma 5, we conclude that \begin{align*} &2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}}\left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k+2}^{\infty} T^{A}_{\Omega , \mu} (f_{j}) \chi_{k}|}{\eta_{1} \sum\limits_{|\gamma|=m}\|D^{\gamma}A\|_{\dot{\wedge}_{\beta}}}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p_{2}(\cdot)} {q_{2}(\cdot)}}} \leq 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}}\left\| \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k+2}^{\infty} T^{A}_{\Omega , \mu} (f_{j}) \chi_{k}|}{\eta_{1} \sum\limits_{|\gamma|=m}\|D^{\gamma}A\|_{\dot{\wedge}_{\beta}}}\right\|^{(q_{2}^{2})_k}_{L^{{p_{2}(\cdot)}}}\\ &\leq C 2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}} \left[ 2^{k\alpha(\cdot)}\sum\limits_{j=k+2}^{\infty}2^{2^{-j(n-(\mu+\beta))}} \left\|\frac{|f_{j}|}{\eta_{1}}\right\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{j}}\|_{L^{p_{1}'(\cdot)}} \|\chi_{B_{k}}\|_{L^{p_{2}(\cdot)}} \right]^{(q^{2}_{2})_k}\\ &\leq C 2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}} \left[ 2^{k\alpha(\cdot)}\sum\limits_{j=k+2}^{\infty}2^{2^{-j(n-(\mu+\beta))}} \left\|\frac{|f_{j}|}{\eta_{1}}\right\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{j}}\|_{L^{p_{1}'(\cdot)}} \|\chi_{B_{k}}\|_{L^{p_{2}(\cdot)}} \right]^{(q^{2}_{2})_k}\\ &\leq C 2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}} \left[ 2^{k\alpha(\cdot)}\sum\limits_{j=k+2}^{\infty}2^{2^{-j(n-(\mu+\beta))}} \left\|\frac{|f_{j}|}{\eta_{1}}\right\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \|\chi_{B_{j}}\|_{L^{p_{1}'(\cdot)}} 2^{-k(\mu+\beta)}\|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}} \right]^{(q^{2}_{2})_k}\end{align*} \begin{align*} &\leq C 2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}} \left[ 2^{k\alpha(\cdot)}\sum\limits_{j=k+2}^{\infty}2^{(j-k)(\mu+\beta)} \left\|\frac{|f_{j}|}{\eta_{1}}\right\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} 2^{-jn}\|\chi_{B_{j}}\|_{L^{p_{1}'(\cdot)}} \|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}} \right]^{(q^{2}_{2})_k}\\ &\leq C 2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}} \left[ 2^{k\alpha(\cdot)}\sum\limits_{j=k+2}^{\infty}2^{(j-k)(\mu+\beta)} \left\|\frac{|f_{j}|}{\eta_{1}}\right\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \frac{\|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}}}{\|\chi_{B_{j}}\|_{L^{p_{1}(\cdot)}}} \right]^{(q^{2}_{2})_k}\\ &\leq C 2^{-k_{0}\lambda_{2}}\sum\limits_{k=0}^{k_{0}} \left[ 2^{k\alpha(\cdot)}\sum\limits_{j=k+2}^{\infty} 2^{(k-j)(n\delta_{2}-(\mu+\beta))}\left\|\frac{|f_{j}|}{\eta_{1}}\right\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \right]^{(q^{2}_{2})_k}, \end{align*} where $$ {(q^{2}_{2})_k}= \left\{\begin{array}{ll} (q_{2})_{-},\qquad \left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=0}^{k-2} T^{A}_{\Omega, \mu} (f_{j}) \chi_{k}|}{\eta_{1} \sum\limits_{|\gamma|=m}\|D^{\gamma}A\|_{\dot{\wedge}_{\beta}}}\right)^{q_{2}(\cdot)}\right\|_{L^{{p_{2}(\cdot)}}} \leq 1, \\ (q_{2})_{+},\qquad \left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=0}^{k-2} T^{A}_{\Omega, \mu} (f_{j}) \chi_{k}|}{\eta_{1} \sum\limits_{|\gamma|=m}\|D^{\gamma}A\|_{\dot{\wedge}_{\beta}}}\right)^{q_{2}(\cdot)}\right\|_{L^{{p_{2}(\cdot)}}} >1. \end{array}\right. $$ Notice that \(f \in {{MK}_{q_{1}(\cdot),~p_{1}(\cdot)}^{\alpha_{+},~\lambda_{1}}(\mathbb{R}^{n})}\), \((\lambda_{2})/(q_{2})_{+}=(\lambda_{1})/(q_{1})_{-}\) and \(\alpha_{+}>(\mu+\beta)- n\delta_{2}+(\lambda_{1})/(q_{1})_{-}\), by the same argument as that of \(\eta_{13}\), we have \begin{align*} & 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{k_{0}}\left\| \left( \frac{2^{k\alpha(\cdot)}|\sum\limits_{j=k+2}^{\infty} T^{A}_{\Omega , \mu} (f_{j}) \chi_{k}|}{\eta_{1} \sum\limits_{|\gamma|=m}\|D^{\gamma}A\|_{\dot{\wedge}_{\beta}}}\right)^{q_{2}(\cdot)}\right\|_{L^{\frac{p_{2}(\cdot)} {q_{2}(\cdot)}}}\\ &\leq C 2^{-k_{0}\lambda_{2}} \sum\limits_{k=0}^{\infty} \left[ \sum\limits_{j=k+2}^{\infty}2^{(k-j)(n\delta_{2}-(\mu+\beta)+\alpha_{+})} \left\|\left(\frac{ 2^{j\alpha_{+}}|f\chi_{k}|}{\eta_{1}}\right)^{q_{1}(\cdot)}\right\|^{\frac{1}{(q^{2}_{1})j}}_{L^\frac{p_{1}(\cdot)} {q_{1}(\cdot)}(\mathbb{R}^{n})} \right]^{(q^{2}_{2})_k}\\ &\leq C \sum\limits_{k=0}^{\infty}2^{(k-k_{0})\lambda_{2}} \left[ \sum\limits_{j=k+2}^{\infty}\ 2^{(k-j)( n\delta_{2}-(\mu+\beta)-(\lambda_{1})/(q_{1})_{-} +\alpha_{+})}\right.\\ &\left. \times \left( 2^{-j\lambda}\sum\limits^{j}_{n=0} \left\|\left(\frac{ 2^{n\alpha_{+}}|f\chi_{n}|}{\eta_{1}}\right)^{q_{1}(\cdot)}\right\|_{L^\frac{p_{1}(\cdot)}{q_{1}(\cdot)}(\mathbb{R}^{n})} \right)^{\frac{1}{(q^{2}_{1})j}} \right]^{(q^{2}_{2})_k}\\ &\leq C, \end{align*} where $$ {(q^{2}_{1})_j}= \left\{\begin{array}{ll} (q_{1})_{-},\qquad \left\| \frac{2^{j\alpha_{+}}| f_{ \chi_{j}}|}{\eta_{1}}\right\|_{L^{{p_{1}(\cdot)}}} \leq 1, \\ (q_{1})_{+},\qquad \left\| \frac{2^{j\alpha_{+}}| f_{ \chi_{j}}|}{\eta_{1}}\right\|_{L^{{p_{1}(\cdot)}}} >1. \end{array}\right. $$ This implies that $$ \eta_{23}\leq C \eta_{1}\sum\limits_{|\gamma|=m}\|D^{\gamma}A\|_{\dot{\wedge}_{\beta}} = \sum\limits_{|\gamma|=m}\|D^{\gamma}A\|_{\dot{\wedge}_{\beta}} \| f\|_{{MK}_{q_{1}(\cdot),p_{1}(\cdot)}^{\alpha(\cdot),\lambda_{1}}(\mathbb{R}^{n})}. $$ Then, the proof of Theorem 2 is finished.

Acknowledgments

The author wishes to express profound gratitude to the reviewers for their useful comments on the manuscript.

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.