By using the boundedness results for the commutators of the fractional integral with variable kernel on variable Lebesgue spaces \(L^{p(\cdot)}(\mathbb{R}^{n})\), the boundedness results are established on variable exponent Herz-Morrey spaces \(M\dot{K}_{q,p(\cdot)}^{\alpha, \lambda}(\mathbb{R}^{n})\).
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 \( 0\leq \mu 0\), \(\chi_{E}\) means its
characteristic function. Now, introduce the definition of the variable exponent Lebesgue spaces.
Definition 1.[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 2.[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 $$ \|f\|_{M\dot{K}_{q,p{(\cdot)}}^{\alpha,\lambda}(\mathbb{R}^{n})}:= \sup_{L\in \mathbb{Z}} 2^{-L\lambda} \left\{\sum_{K=-\infty}^{L} 2^{k\alpha q}\|f\chi_{k}\|_{L^{p(\cdot)}}^{q}\right\}^{1/q}. $$
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) \(\|( b- b_{B_{j}})^{m}\chi_{B_{k}}\|_{L^{p(\cdot)}(\mathbb{R}^{n})} \leq C (k – j)^{m}\|b\|^{m}_{*}\| \chi_{B_{k}}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}.\)
Theorem 15. Assume that \(b^{m} \in BMO(\mathbb{R}^{n})\), \( 0< \mu< n, 0< \beta\leq 1, \lambda < \alpha < n\delta_{1} + \beta , 0< q_{1} \leq q_{2} p_{2}^{+})\), and the integral modulus of continuity \(\omega_{r}(\delta)\) satisfying
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
Theorem 16. Assume that \(b^{m} \in Lip_{\beta}(\mathbb{R}^{n})\), \( 0< \mu< n, 0< \beta\leq 1, \lambda < \alpha < n\delta_{1} + \beta , 0< q_{1} \leq q_{2} p_{2}^{+})\), 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