设$ 0<\alpha<n $, $ S^{n-1} $是$ \mathbb{R}^{n} $中的单位球面, 并且$ \Omega\in L^{s}(\mathbb{S}^{n-1})(s>1) $是零次齐次函数.分数次积分$ T_{\Omega, \mu} $及其极大算子$ M_{\Omega, \mu} $的定义为

$ \begin{equation} T_{\Omega, \mu}f(x) = \int_{\mathbb{R}^{n}}\frac{\Omega(x-y)}{|x-y|^{n-\mu}}f(y)dy, \end{equation} $

(1.1)

$ \begin{equation} M_{\Omega, \mu}f(x) = \sup\limits_{r>0}{\frac{1}{r^{n-\mu}}}\int_{|x-y|<r}|\Omega(x-y)f(y)|dy. \end{equation} $

(1.2)

当$ \mu = 0 $时, $ T_{\Omega, \mu} $与二阶变系数椭圆偏微分方程有着非常密切的联系. 1995年, Caldŕon与Zygmund证明了该算子在变指标Lebesgue空间$ L^{p(\cdot)}(\mathbb{R}^{n}) $上有界.随后, 又有很多学者对其进行了进一步的研究^[1-3].

令$ \gamma = (\gamma_{1}, \gamma_{2}, \cdots, \gamma_{n}) $, 并且$ \gamma_{i}(i = 1, 2, \cdots, n) $是非负整数, 记$ |\gamma| = :\sum\limits_{i = 1}^{n}\gamma_{i} $且

$ \begin{eqnarray*} &&\gamma! = \gamma_{1}!\gamma_{2}!\cdots\gamma_{n}!, \quad x^{\gamma} = x_{1}^{\gamma_{1}}x_{2}^{\gamma_{2}}\cdots x_{n}^{\gamma_{n}}, \\ &&D^{\gamma} = \frac{\partial^{|\gamma|}}{\partial^{\gamma_{1}}x_{1}{\partial^{\gamma_{2}}x_{2}}\cdots{\partial^{\gamma_{n}}x_{n}}}. \end{eqnarray*} $

令$ A(x) $是定义在$ \mathbb{R}^{n} $上的函数, $ R_{m}(A;x, y) $表示定义在$ \mathbb{R}^{n} $上且$ m-1 $阶可导的函数$ A(x) $在点$ x $关于$ y $的$ m $阶Taylor展开式的余项, 即

$ \begin{equation*} R_{m}(A;x, y) = A(x)-\sum\limits_{|\gamma|\leq m-1}\frac{1}{\gamma!}D^{\gamma}A(y)(x-y)^{\gamma}. \end{equation*} $

在2001年, 丁勇^[4]引入了如下一类带粗糙核的多线性分数次积分算子

$ \begin{equation} T_{\Omega, \mu}^{A}f(x) = \int_{\mathbb{R}^{n}}\frac{\Omega(x-y)R_{m}(A;x, y)}{|x-y|^{n-\mu+m-1}}f(y)dy, \end{equation} $

(1.3)

与之相应的分数次极大算子的定义为

$ \begin{equation} M_{\Omega, \mu}^{A}f(x) = \sup\limits_{r>0}\frac{1}{r^{n-\mu+m-1}}\int_{|x-y|<r}|\Omega(x-y)||R_{m}(A;x, y)||f(y)|dy. \end{equation} $

(1.4)

显然, 当$ m = 0 $时, $ T_{\Omega, \mu}^{A} $即为交换子$ [A, T_{\Omega, \mu}] $.当$ m\geq1 $时, $ T_{\Omega, \mu}^{A} $即为上述交换子的非退化推广.丁勇^[4]证明了当$ D^{\gamma}A\in L^{r}(\mathbb{R}^{n})(1\leq r<\infty, |\gamma| = m-1) $时, 该算子在加权Lebesgue空间上有界; Wu和Yang^[5]证明了当$ D^{\gamma}A\in BMO(\mathbb{R}^{n})(|\gamma| = m-1) $时, 该算子在Lebesgue空间上有界.

另一方面, 在1995年, Paluszynski^[6]给出了由Riesz位势算子以及Lipschitz函数生成的广义交换子, 即多线性分数次积分算子, 并给出了Besov空间的一些刻画.在Paluszynski$ ^{[6]} $的启发下, Lu和Zhang^[7]证明了当$ D^{\gamma}A\in \dot{\Lambda}(\mathbb{R}^{n})(|\gamma| = m-1) $时, $ T_{\Omega, \mu}^{A} $在Lebesgue空间上有界.当$ \beta>0 $, 齐次Lipschitz空间的定义为

$ \begin{equation*} \|f\|_{\dot{\bigwedge}_{\beta}} = \sup\limits_{x, h\in\mathbb{R}^{n}, h\neq0}\frac{|\bigtriangleup_{h}^{[\beta]+1}f(x)|}{|h|^{\beta}}<\infty, \end{equation*} $

其中$ \triangle_{h}^{1}f(x) = f(x+h)-f(x), \triangle_{h}^{k+1}f(x) = \triangle_{h}^{k}f(x+h)-\triangle_{h}^{k}f(x), k\geq1 $.

若$ 1\leq q<\infty $, 等价范数为

$ \begin{equation} \|f\|_{\dot{\bigwedge}_{\beta}} = \sup\limits_{Q}\frac{1}{|Q|^{1+\frac{\beta}{n}}}\int_{Q}|f(x)-m_{Q}(f)|dx\\ \approx\sup\limits_{Q}\frac{1}{|Q|^{\frac{\beta}{n}}}(\frac{1}{|Q|}\int_{Q}|f(x)-m_{Q}(f)|^{q}dx)^{\frac{1}{q}}. \end{equation} $

(1.5)

上述都是在一些经典函数空间上的结果, 随着科学的发展, 很多非线性的问题随之而来.这时经典函数空间出现了一定的局限性, 例如它对非标准增长条件下的非线性问题已经失去了效用.在这类非线性问题的研究过程中, 学者越来越多的关注由经典函数空间到变指标函数空间.

令$ p(\cdot):\mathbb{R}^{n}\longrightarrow[1, \infty) $上的可测函数.记变指标Lebesgue空间$ L^{p(\cdot)}(\mathbb{R}^{n}) $为存在某个$ \lambda>0 $使得

$ \int_{\mathbb{R}^{n}}|\frac{f(x)}{\lambda}|^{p(x)}dx<\infty $

成立的$ \mathbb{R}^{n} $上的可测函数$ f $全体.变指标Lebesgue空间$ L^{p(\cdot)}(\mathbb{R}^{n}) $是一个Banach空间, 其范数定义为$ \|f\|_{L^{p(\cdot)}(\mathbb{R}^{n})}: = \text{inf}\{\lambda>0:\rho_{p}(\frac{f}{\lambda})\leq1\}. $记

$ p_{+} = \text{ess}\sup\{p(x):x\in \mathbb{R}^{n}\}, \quad p_{-} = \text{ess}\inf\{p(x):x\in \mathbb{R}^{n}\}. $

用这个符号定义一族变指标

$ \mathcal{P}(\mathbb{R}^{n}): = \{p(\cdot):\mathbb{R}^{n}\rightarrow [1, \infty):p_{-}>1, p_{-}<\infty\}. $

$ p^{'}(\cdot) $是$ p(\cdot) $的共轭指标且$ \frac{1}{p^{'}(\cdot)}+\frac{1}{p(\cdot)} = 1 $.

变指标空间与经典函数空间有很大的不同, 主要是变指标函数空间已经失去了平移不变性, 这一区别导致许多在经典空间中成立的性质在变指标空间中不再成立.随后一些学者发现只要证明Hardy-Littlewood极大算子$ M $在$ L^{p(\cdot)}(\mathbb{R}^{n}) $上有界, 则相应的的经典调和分析和函数空间理论中的许多结论可以在相应的变指标函数空间中成立.

给定函数$ f\in L^{1}_{\text{loc}}(\mathbb{R}^{n}) $, Hardy-Littlewood极大算子$ M $的定义如下

$ \begin{equation*} Mf(x): = \sup\limits_{r>0}r^{-n}\int_{B(x, r)}|f(y)|dy, (x\in\mathbb{R}^{n}), \end{equation*} $

其中$ B(x, r): = \{y\in\mathbb{R}^{n}:|x-y|<r\} $.若$ p(\cdot)\in\mathcal{P}(\mathbb{R}^{n}) $满足下列条件

$ \begin{equation} |p(x)-p(y)|\leq\frac{-C}{log(|x-y|)}, \ \ \ |x-y|\leq\frac{1}{2}, \end{equation} $

(1.6)

$ \begin{equation} |p(x)-p(y)|\leq\frac{C}{log(e+|x|)}, \ \ \ |y|\geq|x|, \end{equation} $

(1.7)

可以得到$ p(\cdot)\in\mathcal{B}(\mathbb{R}^{n}) $.则Hardy-Littlewood极大算子$ M $在$ L^{p(\cdot)}(\mathbb{R}^{n}) $上有界.

在变指标Lebesgue空间$ L^{p(\cdot)}(\mathbb{R}^{n}) $中, 有以下几个重要的命题.

命题1.1^[8] 若$ p(\cdot), p^{'}(\cdot)\in\mathcal{B}(\mathbb{R}^{n}) $, 存在$ 0<r_{1}, r_{2}<1 $且$ C>0 $, 使得

$ \begin{equation} {\frac{\|\chi_{S}\|_{L^{p^{'}(\cdot)}(\mathbb{R}^{n})}}{\|\chi_{B}\|_{L^{p^{'}(\cdot)}(\mathbb{R}^{n})}}}\leq C(\frac{|S|}{|B|})^{r_{1}}, \end{equation} $

(1.8)

$ \begin{equation} {\frac{\|\chi_{S}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}}{\|\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}}}\leq C(\frac{|S|}{|B|})^{r_{2}}, \end{equation} $

(1.9)

其中$ S $是$ \mathbb{R}^{n} $中所有的球$ B $的可测子集.

命题1.2^[8] 若$ p(\cdot)\in\mathcal{B}(\mathbb{R}^{n}) $, 存在常数$ C>0 $, 使得$ \mathbb{R}^{n} $中所有的球$ B $满足

$ \begin{equation*} \frac{1}{|B|}\|\chi_{B}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\|\chi_{B}\|_{L^{p^{'}(\cdot)}(\mathbb{R}^{n})}\leq C. \end{equation*} $

命题1.3^[9] 若$ p(\cdot)\in\mathcal{P}(\mathbb{R}^{n}) $, $ q>p_{+} $且$ {\frac{1}{p(\cdot)} = \frac{1}{\tilde{q}(\cdot)}+\frac{1}{q}} $, 则

$ \begin{equation*} \|fg\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\leq\|f\|_{L^{\tilde{q}(\cdot)}(\mathbb{R}^{n})}\|g\|_{L^{q}(\mathbb{R}^{n})}. \end{equation*} $

命题1.4^[10] 令$ p(\cdot)\in\mathcal{P}(\mathbb{R}^{n}) $满足(1.6)和(1.7)式, 则有

$ \begin{eqnarray*} \|\chi_{Q}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\approx \begin{cases} |Q|^{\frac{1}{p(x)}}, \ \ \ |Q|\leq2^{n}, \\ |Q|^{\frac{1}{p(\infty)}}, \ \ |Q|\geq1, \end{cases} \end{eqnarray*} $

对于所有的方体(球体) $ Q\subset\mathbb{R}^{n} $, 其中$ p(\infty) = \lim\limits_{x\rightarrow\infty}p(x) $.

众所周知, 在研究算子的有界性时, Hölder不等式是一个非常重要的工具, 当然在变指标函数空间也需要同样类似的不等式, 于是就有了广义Hölder不等式.

命题1.5^[11] 若$ p(\cdot)\in\mathcal{P}(\mathbb{R}^{n}) $, 任意的$ f\in{L^{p(\cdot)}(\mathbb{R}^{n})} $, 则$ fg $在$ \mathbb{R}^{n} $上可积并且

$ \begin{equation*} \int_{\mathbb{R}^{n}}|f(x)g(x)|dx\leq r_{p}\|f\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\|g\|_{L^{p^{'}(\cdot)}(\mathbb{R}^{n})}, \end{equation*} $

其中$ r_{p}: = 1+\frac{1}{p_{-}}-\frac{1}{p_{+}} $.上述不等式被称为广义Hölder不等式.

在变指标Lebesgue空间中, Wu和Lan^[6]证明了当$ {D^{\gamma}A}\in{\dot{\Lambda}}_{\beta}(\mathbb{R}^{n}) $时, $ T_{\Omega, \mu}^{A} $是有界的.结果如下

定理1.1^[12] 令$ p(\cdot)\in\mathcal{B}(\mathbb{R}^{n}) $, $ 0<\beta<1 $, $ {D^{\gamma}A}\in{\dot{\Lambda}}_{\beta}(|\gamma| = m-1) $, $ 0<\alpha<\frac{n}{p_{+}} $, $ 0<\alpha+\beta<\frac{n}{p_{+}} $, 并且$ 1<p_{+}<\frac{\alpha+\beta}{n} $.定义$ q(\cdot) $满足$ \frac{1}{p(\cdot)}-\frac{1}{q(\cdot)} = \frac{\alpha+\beta}{n} $, 使得

$ \begin{align*} \|T_{\Omega, \mu}^{A}f\|_{L^{q(\cdot)}(\mathbb{R}^{n})}\leq C{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}}\|f\|_{L^{p(\cdot)}(\mathbb{R}^{n})}. \end{align*} $

2010年, Izuki^[8]引入了变指标Herz空间的定义, 接下来介绍一些定义以及记号.对于任意的$ k\in\mathbb{Z} $, 记$ B_{k} = \{x\in\mathbb{R}^{n}:|x|\leq2^{k}\} $和$ R_{k} = B_{k}\backslash B_{k-1} $; 对任意的$ k\in\mathbb{Z} $, 记$ \chi_{k} = \chi_{R_{k}} $; 对任意的$ k\in\mathbb{N}_{0} $, 记$ \tilde{\chi_{k}} = \chi_{k} $, 其中当$ k\in\mathbb{N} $时, 有$ \tilde{\chi_{k}} = \chi_{k} $; 当$ k = 0 $时, 有$ \tilde{\chi_{0}} = \chi_{B_{0}} $.

定义1.1^[8] 令$ \alpha\in\mathbb{R} $, $ 0<q\leq\infty $且$ p(\cdot)\in\mathcal{P}(\mathbb{R}^{n}) $.齐次变指标$ \dot{K}^{\alpha, q}_{p(\cdot)}(\mathbb{R}^{n}) $空间定义为

$ \begin{equation*} \dot{K}^{\alpha, q}_{p(\cdot)}(\mathbb{R}^{n}): = \{f\in{L_{loc}^{p(\cdot)}(\mathbb{R}^{n}\setminus\{0\})}:\|f\|_{\dot{K}^{\alpha, q}_{p(\cdot)}(\mathbb{R}^{n})}<\infty\}, \end{equation*} $

其中$ \|f\|_{\dot{K}^{\alpha, q}_{p(\cdot)}(\mathbb{R}^{n})}: = \|\{2^{\alpha l}\|f\chi_{l}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}\}_{l = -\infty}^{\infty}\|_{\ell^{q}(\mathbb{Z})} $.

本文将Wu和Lan^[12]的结果推广到变指标Herz空间, 建立多线性分数次积分算子在变指标Herz空间上的有界性.

定理1.2  令$ p_{1}(\cdot), p_{2}(\cdot)\in\mathcal{B}(\mathbb{R}^{n}) $, $ 0<\beta<1 $, $ {D^{\gamma}A}\in{\dot{\Lambda}}_{\beta}(|\gamma| = m-1) $, $ s>(p_{1}^{'})_{+} $, $ 0<r_{1}, r_{2}<\infty $且满足命题1.1和(1.8)式, $ nr_{2}+\beta+\mu<\alpha<nr_{1}-(\beta+\mu+\frac{n-1}{s}) $.如果$ \frac{1}{p_{1}(\cdot)}-\frac{1}{p_{2}(\cdot)} = \frac{\beta+\mu}{n} $, 且$ 0<q_{1}<q_{2}<\infty $使得

$ \begin{align*} \|T_{\Omega, \mu}^{A}f\|_{\dot{K}^{\alpha, q_{2}}_{p_{2}(\cdot)}(\mathbb{R}^{n})}\leq C{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}}\|f\|_{\dot{K}^{\alpha, q_{1}}_{p_{1}(\cdot)}(\mathbb{R}^{n})}. \end{align*} $

接下来介绍比变指标Herz空间更广泛的变指标Herz-Morrey空间, 并在此空间上建立相应的结论.

定义1.2^[13] 令$ \alpha\in\mathbb{R} $, $ 0<q\leq\infty $, $ p(\cdot)\in\mathcal{P}(\mathbb{R}^{n}) $且$ 0\leq\lambda<\infty $.齐次变指标Herz-Morrey空间$ M\dot{K}_{q, p(\cdot)}^{\alpha, \lambda}(\mathbb{R}^{n}) $的定义如下

$ \begin{equation*} M\dot{K}_{q, p(\cdot)}^{\alpha, \lambda}(\mathbb{R}^{n}): = \{f\in{L_{loc}^{p(\cdot)}((\mathbb{R}^{n}\setminus\{0\})}:\|f\|_{M\dot{K}_{q, p(\cdot)}^{\alpha, \lambda} (\mathbb{R}^{n}))}<\infty\}, \end{equation*} $

其中$ \|f\|_{M\dot{K}_{q, p(\cdot)}^{\alpha, \lambda}(\mathbb{R}^{n})}: = \sup\limits_{L\in\mathbb{Z}} 2^{-L\lambda}(\sum\limits^{L}_{k = -\infty}2^{\alpha q k}\|f\chi_{k}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}^{q})^{\frac{1}{q}} $.显然, 当$ \lambda = 0 $时, $ M\dot{K}_{q, p(\cdot)}^{\alpha, \lambda}(\mathbb{R}^{n}) = \dot{K}_{p(\cdot)}^{\alpha, q}(\mathbb{R}^{n}). $

定理1.3 令$ p_{1}(\cdot), p_{2}(\cdot)\in\mathcal{B}(\mathbb{R}^{n}) $, $ 0<\beta<1 $, $ {D^{\gamma}A}\in{\dot{\Lambda}}_{\beta}(|\gamma| = m-1) $. $ s>(p_{1}^{'})_{+} $, $ 0<\lambda<\infty $, $ 0<r_{1}, r_{2}<\infty $且满足(1.6)和(1.7)式, $ nr_{2}+\beta+\mu+\lambda<\alpha<nr_{1}-(\beta+\mu+\frac{n-1}{s}) $.如果$ \frac{1}{p_{1}(\cdot)}-\frac{1}{p_{2}(\cdot)} = \frac{\beta+\mu}{n} $, 且$ 0<q_{1}<q_{2}<\infty $使得

$ \begin{align*} \|T_{\Omega, \mu}^{A}f\|_{M\dot{K}^{\alpha, \lambda}_{q_{2}, p_{2}(\cdot)}(\mathbb{R}^{n})}\leq C{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}}\|f\|_{M\dot{K}^{\alpha, \lambda}_{q_{1}, p_{1}(\cdot)}(\mathbb{R}^{n})}. \end{align*} $

定理1.4 在定理1.2或定理1.3的条件下, 多线性分数次极大算子$ M_{\Omega, \mu}^{A} $在$ \dot{K}^{\alpha, q}_{p(\cdot)}(\mathbb{R}^{n}) $, $ M\dot{K}_{q, p(\cdot)}^{\alpha, \lambda}(\mathbb{R}^{n}) $有界.

注 当$ m = 0 $时, $ T_{\Omega, \mu}^{A}f(x) $即为交换子$ [A, T_{\Omega, \mu}f(x)] $在$ \dot{K}^{\alpha, q}_{p(\cdot)}(\mathbb{R}^{n}) $以及$ M\dot{K}_{q, p(\cdot)}^{\alpha, \lambda}(\mathbb{R}^{n}) $有界.

全文中, $ C $表示一个不依赖于主要参数的常数，但其值在不同的地方可能不尽相同.

在证明定理1.2和定理1.3之前, 需要一些必要的引理.

引理2.1^[7] $ A(x) $是$ \mathbb{R}^{n} $上的一个函数, 且$ A(x)\in{L_{\text{loc}}^{q}(\mathbb{R}^{n})}\; (q>n), $

$ \begin{equation*} |R_{m}(A;x, y)|\leq C|x-y|^{m}\sum\limits_{|\gamma| = m}(\frac{1}{|Q_{x}^{y}|}\int_{Q_{x}^{y}}|D^{\gamma}A(z)|^{q}dy)^{\frac{1}{q}}, \end{equation*} $

其中$ Q_{x}^{y} $是以$ x $为心, 以$ 4\sqrt{n}|x-y| $为边长的方体.

引理2.2^[7] 令$ Q^{\ast}\subset Q, g\in{{\dot{\Lambda}}_{\beta}(\mathbb{R}^{n})}(0<\beta<1) $, 则$ |m_{Q^{\ast}}(g)-m_{Q}(g)|\leq C|Q|^{\frac{\beta}{n}}\|g\|_{\dot{\bigwedge}_{\beta}}. $

引理2.3^[14] 设$ \Omega(x-y)\in{L^{s}(\mathbb{S}^{n-1})}(s>\frac{n}{(n-\alpha)}), k, j\in\mathbb{Z} $, 则

(1) 当$ k\leq{j-3}, \; x\in{R_{k}} $时, $ ( \int_{R_{j}}|\Omega(x-y)|^{s}dy)^{\frac{1}{s}}\leq C 2^{\frac{jn}{s}}\|\Omega\|_{L^{s}(\mathbb{S}^{n-1})} $.

(2) 当$ k\geq{j+3}, \; x\in{R_{k}} $时, $ ( \int_{R_{j}}|\Omega(x-y)|^{s}dy)^{\frac{1}{s}}\leq C 2^{\frac{(j-k+kn)}{s}}\|\Omega\|_{L^{s}(\mathbb{S}^{n-1})} $.

应用以上结果, 首先证明定理1.2.

定理1.2的证明 令$ f\in\dot{K}^{\alpha, q_{1}}_{p_{1}(\cdot)}(\mathbb{R}^{n}) $, 因为$ 0<q_{1}/{q_{2}}\leq1 $, 由不等式

$ \begin{align} ({\sum\limits_{h = 1}^{\infty}}a_{h})^{q_{1}/{q_{2}}}\leq{\sum\limits_{h = 1}^{\infty}}(a_{h})^{q_{1}/{q_{2}}} (a_{1}, a_{2}, a_{3}, \cdot\cdot\cdot>0). \end{align} $

(2.1)

对于任意$ j\in\mathbb{Z} $, 记$ f_{j}: = f \chi_{j} $, 则$ f = {\sum\limits_{j = -\infty}^{\infty}f_{j}} $.运用()式可得

$ \begin{align*} \|T_{\Omega, \mu}^{A}f\|_{\dot{K}^{\alpha, q_{2}}_{p_{2}(\cdot)}(\mathbb{R}^{n})}^{q_{1}} \leq&{\sum\limits_{k = -\infty}^{\infty}}2^{\alpha q_{1} k}({\sum\limits_{j = -\infty}^{k-3}}\|(T_{\Omega, \mu}^{A}f_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})})^{q_{1}}\\ &+{\sum\limits_{k = -\infty}^{\infty}}2^{\alpha q_{1} k}({\sum\limits_{j = k-2}^{k+2}}\|(T_{\Omega, \mu}^{A}f_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})})^{q_{1}}\\ &+{\sum\limits_{k = -\infty}^{\infty}}2^{\alpha q_{1} k}({\sum\limits_{j = k+3}^{\infty}}\|(T_{\Omega, \mu}^{A}f_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})})^{q_{1}}\\ = :&\rm I_{1}+I_{2}+I_{3}. \end{align*} $

首先估计$ {\rm I}_{2} $, 当$ D^{\gamma}A\in{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})} $, 由定理1.1中$ T_{\Omega, \mu}^{A} $的$ (L^{p_{1}(\cdot)}(\mathbb{R}^{n}), L^{p_{2}(\cdot)}(\mathbb{R}^{n})) $有界性可知

$ \begin{align*} {\rm I_{2}}\leq& C{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}^{q_{1}}}{\sum\limits_{k = -\infty}^{\infty}}({\sum\limits_{j = k-2}^{k+2}}\|f_{j}\|_{L^{p_{1}(\cdot)} (\mathbb{R}^{n})}2^{\alpha j} 2^{\alpha(k-j)})^{q_{1}}\\ \end{align*} $

$ \begin{align*} \leq& C{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}^{q_{1}}}\|f\|^{q_{1}}_{\dot{K}^{\alpha, q_{1}}_{p_{1}(\cdot)}(\mathbb{R}^{n})}. \end{align*} $

接下来估计$ \rm I_{1} $, 为此由(1.3)先对$ |R_{m}(A;x, y)| $进行估计.

对于任意$ x\in\mathbb{R}^{n} $, 且$ Q $是以$ x $为心, $ 2\sqrt{n}|x-y| $为边长的方体, 令$ A_{Q}(y) = A(y)-\sum\limits_{|\gamma| = m-1}\frac{1}{\gamma!}m_{Q}(D^{\gamma}A)\cdot y^{\gamma} $, 显然可得$ R_{m}(A;x, y) = R_{m}(A_{Q};x, y) $, 其中$ m_{Q}(D^{\gamma}A) $记作$ D^{\gamma}A $在Q上的平均.再运用引理2.1, 对于任意的$ y $,

$ \begin{align} |R_{m}(A;x, y)|\leq&|R_{m-1}(A_{Q};x, y)|+C\sum\limits_{|\gamma| = m-1}|D^{\gamma}A_{Q}(x)||x-y|^{m-1}\\ \leq& C|x-y|^{m-1}\sum\limits_{|\gamma| = m-1}(\frac{1}{|Q_{x}^{y}|}\int_{Q_{x}^{y}}|D^{\gamma}A_{Q}(z)|^{q}dy)^{\frac{1}{q}}\\ &+C\sum\limits_{|\gamma| = m-1}|D^{\gamma}A_{Q}(x)||x-y|^{m-1}, \end{align} $

(2.2)

其中$ Q_{x}^{y} $是以$ x $为心, 以$ 4\sqrt{n}|x-y| $为边长的方体.显然$ Q_{x}^{y}\subset3Q $, 由引理2.2可得

$ \begin{align} (\frac{1}{|Q_{x}^{y}|}\int_{Q_{x}^{y}}|D^{\gamma}A_{Q}(z)|^{q}dy)^{\frac{1}{q}} = &(\frac{1}{|Q_{x}^{y}|}\int_{Q_{x}^{y}}|D^{\gamma}A(z)-m_{Q}(D^{\gamma}A)|^{q}dy)^{\frac{1}{q}}\\ \leq&(\frac{1}{|Q_{x}^{y}|}\int_{Q_{x}^{y}}|D^{\gamma}A(z)-m_{Q_{x}^{y}}(D^{\gamma}A)|^{q}dy)^{\frac{1}{q}}\\ &+|m_{Q_{x}^{y}}(D^{\gamma}A)-m_{3Q}(D^{\gamma}A)|+|m_{3Q}(D^{\gamma}A)-m_{Q}(D^{\gamma}A)|\\ \leq& C{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}}|x-y|^{\beta}. \end{align} $

(2.3)

再次运用引理2.2可得

$ \begin{align} |D^{\gamma}A_{Q}(x)|\leq C{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}}|Q|^{\frac{\beta}{n}} \leq C{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}}|x-y|^{\beta}. \end{align} $

(2.4)

联立(2.2)–(2.4)式可得

$ \begin{align} |R_{m}(A;x, y)|\leq C\sum\limits_{|\gamma| = m-1}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}}|x-y|^{m-1+\beta}. \end{align} $

(2.5)

对于任意的$ {j, k}\in\mathbb{Z} $且$ j\leq{k-3} $, 运用()式和广义Hölder不等式, 可以得到

$ \begin{align} |(T_{\Omega, \mu}^{A}f_{j})\chi_{k}| &\leq\int_{R_{j}}{\frac{|\Omega(x-y)||R_{m}(A;x, y)|}{|x-y|^{n-\mu+m-1}}|f_{j}(y)|dy\cdot\chi_{k}}\\ &\leq C {2^{k}}^{(\beta+\mu-n)}{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}} \int_{R^{n}}|\Omega(x-y)||f_{j}(y)|\chi_{j}dy\cdot\chi_{k}\\ &\leq C {2^{k}}^{(\beta+\mu-n)}{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}}{\|f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}} {\|\Omega(x-y)\chi_{j}\|_{L^{p_{1}^{'}(\cdot)}(\mathbb{R}^{n})}}\cdot\chi_{k}. \end{align} $

(2.6)

因为$ s>(p_{1}^{'})_{+} $且$ {\frac{1}{p_{1}^{'}(\cdot)} = \frac{1}{\tilde{p}_{1}^{'}(\cdot)}+\frac{1}{s}} $, 运用命题1.3和引理2.3可得

$ \begin{align} \|\Omega(x-y)\chi_{j}\|_{L^{p_{1}^{'}(\cdot)}(\mathbb{R}^{n})} &\leq\|\Omega(x-y)\chi_{j}\|_{L^{s}(\mathbb{R}^{n})}\|\chi_{j}\|_{L^{\tilde{p}_{1}^{'}(\cdot)}(\mathbb{R}^{n})}\\ &\leq C 2^{\frac{(j-k+kn)}{s}}\|\chi_{B_{j}}\|_{L^{\tilde{p}_{1}^{'}(\cdot)}(\mathbb{R}^{n})}. \end{align} $

(2.7)

由命题1.4, 当$ |B_{j}|\leq2^{n} $且$ x_{j}\in B_{j} $, 有

$ \begin{align*} \|\chi_{B_{j}}\|_{L^{\tilde{p}_{1}^{'}(\cdot)}(\mathbb{R}^{n})}\approx|B_{j}|^{\frac{1}{\tilde{p}_{1}^{'}(x_{j})}} \approx\|\chi_{B_{j}}\|_{L^{p_{1}^{'}(\cdot)}(\mathbb{R}^{n})}|B_{j}|^{-\frac{1}{s}}. \end{align*} $

当$ |B_{j}|\geq1 $, 有

$ \begin{align*} \|\chi_{B_{j}}\|_{L^{\tilde{p}_{1}^{'}(\cdot)}(\mathbb{R}^{n})}\approx|B_{j}|^{\frac{1}{\tilde{p}_{1}^{'}(\infty)}} \approx\|\chi_{B_{j}}\|_{L^{p_{1}^{'}(\cdot)}(\mathbb{R}^{n})}|B_{j}|^{-\frac{1}{s}}. \end{align*} $

故有

$ \begin{align} \|\chi_{B_{j}}\|_{L^{\tilde{p}_{1}^{'}(\cdot)}(\mathbb{R}^{n})}\approx\|\chi_{B_{j}}\|_{L^{p_{1}^{'}(\cdot)}(\mathbb{R}^{n})}|B_{j}|^{-\frac{1}{s}}. \end{align} $

(2.8)

联立(2.6)–(2.8)式可得

$ \begin{align} &\|(T_{\Omega, \mu}^{A}f_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\\ \leq& C {2^{k}}^{(\beta+\mu-n)}{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}}{\|f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}} {\|\Omega(x-y)\chi_{j}\|_{L^{p_{1}^{'}(\cdot)}(\mathbb{R}^{n})}}\|\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\\ \leq& C{2^{k}}^{(\beta+\mu-n)}2^{\frac{(k-j)(n-1)}{s}}{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}} {\|f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}}\|\chi_{B_{j}}\|_{L^{p_{1}^{'}(\cdot)}(\mathbb{R}^{n})}\|\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}. \end{align} $

(2.9)

运用命题1.2和(1.1)式可得

$ \begin{align} 2^{k(\beta+\mu-n)}\|\chi_{B_{j}}\|_{L^{p_{1}^{'}(\cdot)}(\mathbb{R}^{n})}\|\chi_{B_{k}}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} &\leq C2^{k(\beta+\mu)}\|\chi_{B_{j}}\|_{L^{p_{1}^{'}(\cdot)}(\mathbb{R}^{n})}\|\chi_{B_{k}}\|^{-1}_{L^{p_{2}^{'}(\cdot)}(\mathbb{R}^{n})}\\ &\leq C2^{k(\beta+\mu)}\|\chi_{B_{j}}\|_{L^{p_{1}^{'}(\cdot)}(\mathbb{R}^{n})}\|\chi_{B_{j}}\|^{-1}_{L^{p_{2}^{'}(\cdot)}(\mathbb{R}^{n})}2^{nr_{1}(j-k)}. \end{align} $

(2.10)

另一方面, 由分数次积分$ I^{\beta+\mu} $可以得到

$ \begin{align} I^{\beta+\mu}(\chi_{B_{j}})(x)\geq I^{\beta+\mu}(\chi_{B_{j}})(x)\chi_{B_{j}}\geq C2^{j(\beta+\mu)}\chi_{B_{j}}. \end{align} $

(2.11)

接下来, 运用Hölder不等式, (2.11)式, $ I^{\beta+\mu} $的$ (L^{p_{1}(\cdot)}(\mathbb{R}^{n}), L^{p_{2}(\cdot)}(\mathbb{R}^{n})) $有界性以及命题1.2可知

$ \begin{align} \|\chi_{B_{j}}\|^{-1}_{L^{p_{2}^{'}(\cdot)}(\mathbb{R}^{n})}&\leq C2^{-nj}\|\chi_{B_{j}}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})} \leq C 2^{-j(\beta+\mu)}2^{-nj}\|I^{\beta+\mu}(\chi_{B_{j}})\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\\ &\leq C 2^{-j(\beta+\mu)}2^{-nj}\|\chi_{B_{j}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} \leq C 2^{-j(\beta+\mu)}\|\chi_{B_{j}}\|^{-1}_{L^{p_{1}^{'}(\cdot)}(\mathbb{R}^{n})}. \end{align} $

(2.12)

联立(2.9)–(2.12)式可知

$ \begin{align} {\rm {I_{1}}}\leq C {\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}^{q_{1}}}{\sum\limits_{k = -\infty}^{\infty}}({\sum\limits_{j = -\infty}^{k-3}} \|f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}2^{\alpha j} 2^{e_{1}(k-j)})^{q_{1}}, \end{align} $

(2.13)

其中$ e_{1}: = \beta+\mu-nr_{1}+\alpha+\frac{n-1}{s}<0 $.为了继续估计(2.13)式.考虑以下这两种情形$ 1<q_{1}<\infty $和$ 0<q_{1}\leq1 $.

情形1 若$ 1<q_{1}<\infty $, 运用Hölder不等式, 可得

$ \begin{align*} \rm I_{1}&\leq C{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}^{q_{1}}} {\sum\limits_{k = -\infty}^{\infty}}{\sum\limits_{j = -\infty}^{k-3}}2^{\alpha jq_{1}}\|f_{j}\|^{q_{1}}_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} 2^{e_{1}(k-j)\frac{q_{1}}{2}} \times({\sum\limits_{j = -\infty}^{k-3}}2^{e_{1}(k-j)\frac{q^{'}_{1}}{2}})^{q_{1}/q^{'}_{1}}\\ &\leq C{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}^{q_{1}}}{\sum\limits_{j = -\infty}^{\infty}}2^{\alpha jq_{1}}\|f_{j}\|^{q_{1}}_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}{\sum\limits_{k = j+3}^{\infty}}2^{e_{1}(k-j)\frac{q_{1}}{2}}\\ &\leq C{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}^{q_{1}}} \|f\|^{q_{1}}_{\dot{K}^{\alpha, q_{1}}_{p_{1}(\cdot)}(\mathbb{R}^{n})}. \end{align*} $

情形2 若$ 0<q_{1}\leq1 $, 运用(2.1)式, 用$ q_{1} $代替$ q_{1}/q_{2} $可得

$ \begin{align*} {\rm I_{1}}&\leq C{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}^{q_{1}}}{\sum\limits_{k = -\infty}^{\infty}}{\sum\limits_{j = -\infty}^{k-3}}\|f_{j}\|^{q_{1}}_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}2^{\alpha jq_{1}} 2^{e_{1}(k-j)q_{1}}\\ &\leq C{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}^{q_{1}}}\|f\|^{q_{1}}_{\dot{K}^{\alpha, q_{1}}_{p_{1}(\cdot)}(\mathbb{R}^{n})}. \end{align*} $

最后估计$ {\rm I}_{3} $, 对于任意的$ {j, k}\in\mathbb{Z} $且$ j\geq{k+3} $, 类似于(2.9)式的估计, 可以得到

$ \begin{align} &\|(T_{\Omega, \mu}^{A}f_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\\ \leq& C2^{j(\beta+\mu-n)}{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}}{\|f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}} \|\chi_{B_{j}}\|_{L^{p_{1}^{'}(\cdot)}(\mathbb{R}^{n})}\|\chi_{B_{k}}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}. \end{align} $

(2.14)

运用命题1.2和(1.8)式, 类似于(2.10)式的估计可得

$ \begin{align} &2^{j(\beta+\mu-n)}\|\chi_{B_{j}}\|_{L^{p_{1}^{'}(\cdot)}(\mathbb{R}^{n})}\|\chi_{B_{k}}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}\\ \leq& C2^{j(\beta+\mu)}\|\chi_{B_{k}}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\|\chi_{B_{k}}\|^{-1}_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}2^{nr_{2}(j-k)}. \end{align} $

(2.15)

另一方面, 由分数次积分$ I^{\beta+\mu} $可以知道

$ \begin{align} I^{\beta+\mu}(\chi_{B_{k}})(x)\geq I^{\beta+\mu}(\chi_{B_{k}})(x)\chi_{B_{k}}\geq C2^{k(\beta+\mu)}\chi_{B_{k}}. \end{align} $

(2.16)

类似于(2.11)式的估计, 可以得到

$ \begin{align} \|\chi_{B_{k}}\|^{-1}_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}&\leq C 2^{-nk}\|\chi_{B_{k}}\|_{L^{p^{'}_{1}(\cdot)}(\mathbb{R}^{n})}\leq C 2^{-k(\beta+\mu)}\|\chi_{B_{k}}\|^{-1}_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})}. \end{align} $

(2.17)

联立(2.14) – (2.18)式可得

$ \begin{align} {\rm I_{3}}\leq C{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}^{q_{1}}}{\sum\limits_{k = -\infty}^{\infty}}({\sum\limits_{j = k+3}^{\infty}} \|f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}2^{\alpha j} 2^{e_{2}(j-k)})^{q_{1}}, \end{align} $

(2.18)

其中$ e_{2}: = \beta+\mu+nr_{2}-\alpha<0 $.为了继续估计$ (2.18) $式, 考虑以下这两种情形$ 1<q_{1}<\infty $和$ 0<q_{1}\leq1 $.

情形1 若$ 1<q_{1}<\infty $, 运用Hölder不等式, 可以得到

$ \begin{align*} \rm I_{3}&\leq C{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}^{q_{1}}}{\sum\limits_{k = -\infty}^{\infty}}{\sum\limits_{j = k+3}^{\infty}}2^{\alpha jq_{1}}\|f_{j}\|^{q_{1}}_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})} 2^{e_{2}(j-k)\frac{q_{1}}{2}} \times({\sum\limits_{j = k+3}^{\infty}}2^{e_{2}(j-k)\frac{q^{'}_{1}}{2}})^{q_{1}/q^{'}_{1}}\\ &\leq C{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}^{q_{1}}}{\sum\limits_{j = -\infty}^{\infty}}2^{\alpha jq_{1}}\|f_{j}\|^{q_{1}}_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}{\sum\limits_{k = j-3}^{\infty}}2^{e_{2}(j-k)\frac{q_{1}}{2}}\\ &\leq C{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}^{q_{1}}} \|f\|^{q_{1}}_{\dot{K}^{\alpha, q_{1}}_{p_{1}(\cdot)}(\mathbb{R}^{n})}. \end{align*} $

情形2 若$ 0<q_{1}\leq1 $, 用$ () $式, 用$ q_{1} $代替$ q_{1}/q_{2} $可得

$ \begin{align*} {\rm I_{3}}\leq& C{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}^{q_{1}}}{\sum\limits_{k = -\infty}^{\infty}}{\sum\limits_{j = k+3}^{\infty}} \|f_{j}\|^{q_{1}}_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}2^{\alpha jq_{1}} 2^{e_{2}(j-k)q_{1}}\\ \leq& C{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}^{q_{1}}} \|f\|^{q_{1}}_{\dot{K}^{\alpha, q_{1}}_{p_{1}(\cdot)}(\mathbb{R}^{n})}. \end{align*} $

这样就完成了定理1.2的证明.接下来证明定理1.3.

定理1.3的证明 令$ f\in M\dot{K}^{\alpha, \lambda}_{q_{1}, p_{1}(\cdot)} $, 记$ f_{j}: = f \chi_{j} $($ j\in\mathbb{Z} $), 则$ f = {\sum\limits_{j = -\infty}^{\infty}f_{j}} $, 用(2.1)式可得

$ \begin{align*} \|T_{\Omega, \mu}^{A}f\|_{M\dot{K}_{q_{2}, p_{2}(\cdot)}^{\alpha, \lambda}(\mathbb{R}^{n})}^{q_{1}} \leq& C\sup\limits_{L\in\mathbb{Z}}2^{-L\lambda q_{1}}{\sum\limits_{k = -\infty}^{L}}2^{\alpha q_{1} k}({\sum\limits_{j = -\infty}^{k-3}}\|(T_{\Omega, \mu}^{A}f_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})})^{q_{1}}\\ &+C\sup\limits_{L\in\mathbb{Z}}2^{-L\lambda q_{1}}\sum\limits_{k = -\infty}^{L}2^{\alpha q_{1} k}({\sum\limits_{j = k-2}^{k+2}}\|(T_{\Omega, \mu}^{A}f_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})})^{q_{1}}\\ &+C\sup\limits_{L\in\mathbb{Z}}2^{-L\lambda q_{1}}{\sum\limits_{k = -\infty}^{L}}2^{\alpha q_{1} k}({\sum\limits_{j = k+3}^{\infty}}\|(T_{\Omega, \mu}^{A}f_{j})\chi_{k}\|_{L^{p_{2}(\cdot)}(\mathbb{R}^{n})})^{q_{1}}\\ = :&\rm D_{1}+D_{2}+D_{3}. \end{align*} $

首先估计$ {\rm D}_{2} $, 当$ D^{\gamma}A\in{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})} $时, 由定理1.1中$ T_{\Omega, \mu}^{A} $的$ (L^{p_{1}(\cdot)}(\mathbb{R}^{n}), L^{p_{2}(\cdot)}(\mathbb{R}^{n})) $有界性可知

$ \begin{align*} {\rm D_{2}}\leq& C{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}^{q_{1}}}\sup\limits_{L\in\mathbb{Z}}2^{-L\lambda q_{1}}\sum\limits_{k = -\infty}^{L}({\sum\limits_{j = k-2}^{k+2}}\|f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}2^{\alpha j} 2^{\alpha(k-j)})^{q_{1}}\\ \leq &C{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}^{q_{1}}} \|f\|^{q_{1}}_{M\dot{K}^{\alpha, \lambda}_{q_{1}, p_{1}(\cdot)}(\mathbb{R}^{n})}. \end{align*} $

类似于定理1.2中$ {\rm I}_{1} $的估计方法, 可得

$ \begin{align} {\rm D_{1}}\leq C{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}^{q_{1}}}\sup\limits_{L\in\mathbb{Z}}2^{-L\lambda q_{1}}{\sum\limits_{k = -\infty}^{L}}({\sum\limits_{j = -\infty}^{k-3}}\|f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}2^{\alpha j} 2^{d_{1}(k-j)})^{q_{1}}. \end{align} $

(2.19)

其中$ d_{1}: = \beta+\mu-nr_{1}+\alpha+\frac{n-1}{s}<0 $, 为了继续估计$ (2.19) $式, 考虑以下这两种情形$ 1<q_{1}<\infty $和$ 0<q_{1}\leq1 $.

情形1 若$ 1<q_{1}<\infty $, 运用Hölder不等式, 可以得到

$ \begin{align*} \rm D_{1} \leq& C{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}^{q_{1}}}\sup\limits_{L\in\mathbb{Z}}2^{-L\lambda q_{1}}{\sum\limits_{k = -\infty}^{L}}{\sum\limits_{j = -\infty}^{k-3}}2^{\alpha jq_{1}}\|f_{j}\|^{q_{1}}_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}2^{d_{1}(k-j)\frac{q_{1}}{2}}\\ &\times({\sum\limits_{j = -\infty}^{k-3}}2^{d_{1}(k-j)\frac{q^{'}_{1}}{2}})^{q_{1}/q^{'}_{1}}\\ \leq& C{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}^{q_{1}}} \|f\|^{q_{1}}_{M\dot{K}^{\alpha, \lambda}_{q_{1}, p_{1}(\cdot)}(\mathbb{R}^{n})}. \end{align*} $

情形2 若$ 0<q_{1}\leq1 $, 运用(2.1)式, 用$ q_{1} $代替$ q_{1}/q_{2} $可得

$ \begin{align*} \rm D_{1}&\leq C{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}^{q_{1}}}\sup\limits_{L\in\mathbb{Z}}2^{-L\lambda q_{1}}{\sum\limits_{k = -\infty}^{L}}{\sum\limits_{j = -\infty}^{k-3}}\|f_{j}\|^{q_{1}}_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}2^{\alpha jq_{1}} 2^{d_{1}(k-j)q_{1}}\\ &\leq C{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}^{q_{1}}}\|f\|^{q_{1}}_{M\dot{K}^{\alpha, \lambda}_{q_{1}, p_{1}(\cdot)}(\mathbb{R}^{n})}. \end{align*} $

最后估计$ {\rm D}_{3} $,

$ \begin{align} {\rm D_{3}}\leq C{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}^{q_{1}}}\sup\limits_{L\in\mathbb{Z}}2^{-L\lambda q_{1}}{\sum\limits_{k = -\infty}^{L}}({\sum\limits_{j = k+3}^{\infty}}\|f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}2^{\alpha j} 2^{d_{2}(j-k)})^{q_{1}}, \end{align} $

(2.20)

其中$ d_{2}: = \beta+\mu+nr_{2}-\alpha<0 $.为了继续估计$ (2.20) $式, 考虑以下这两种情形$ 1<q_{1}<\infty $和$ 0<q_{1}\leq1 $.

情形1 若$ 1<q_{1}<\infty $, 有

$ \begin{align*} {\rm D_{3}}\leq& C{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}^{q_{1}}}\sup\limits_{L\in\mathbb{Z}}2^{-L\lambda q_{1}}{\sum\limits_{k = -\infty}^{L}}({\sum\limits_{j = k+3}^{L}}\|f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}2^{\alpha j} 2^{d_{2}(j-k)})^{q_{1}}\\ &+C{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}^{q_{1}}}\sup\limits_{L\in\mathbb{Z}}2^{-L\lambda q_{1}}{\sum\limits_{k = -\infty}^{L}}({\sum\limits_{j = L+1}^{\infty}}\|f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}2^{\alpha j} 2^{d_{2}(j-k)})^{q_{1}}\\ = :&\rm D_{31}+D_{32}. \end{align*} $

对于$ {\rm D}_{31} $, 运用Hölder不等式, 得到

$ \begin{align*} \rm D_{31} \leq& C{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}^{q_{1}}}\sup\limits_{L\in\mathbb{Z}}2^{-L\lambda q_{1}}{\sum\limits_{k = -\infty}^{L}}{\sum\limits_{j = k+3}^{L}}2^{\alpha jq_{1}}\|f_{j}\|^{q_{1}}_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}2^{d_{2}(j-k)\frac{q_{1}}{2}}\\ &\times({\sum\limits_{j = -\infty}^{k+3}}2^{d_{2}(j-k)\frac{q^{'}_{1}}{2}})^{q_{1}/q^{'}_{1}}\\ \leq& C{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}^{q_{1}}} \|f\|^{q_{1}}_{M\dot{K}^{\alpha, \lambda}_{q_{1}, p_{1}(\cdot)}(\mathbb{R}^{n})}. \end{align*} $

对于$ {\rm D}_{32} $, 由已知条件$ d_{2}+\lambda<0 $和Hölder不等式, 可得

$ \begin{align*} \rm D_{32} \leq& C{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}^{q_{1}}}\sup\limits_{L\in\mathbb{Z}}2^{-L\lambda q_{1}}{\sum\limits_{k = -\infty}^{L}}({\sum\limits_{j = L+1}^{\infty}}\|f_{j}\|_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}2^{\alpha j} 2^{(j-k)\frac{(d_{2}+\lambda)}{2}}2^{(j-k)\frac{(d_{2}-\lambda)}{2}})^{q_{1}}\\ \leq& C{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}^{q_{1}}}\sup\limits_{L\in\mathbb{Z}}2^{-L\lambda q_{1}}{\sum\limits_{k = -\infty}^{L}}{\sum\limits_{j = L+1}^{\infty}}2^{\lambda jq_{1}}2^{(j-k)(d_{2}+\lambda)\frac{q_{1}}{2}}\\ &\times2^{-\lambda jq_{1}}\sum\limits_{m = -\infty}^{j}2^{\alpha mq_{1}}\|f_{m}\|^{q_{1}}_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\\ \leq& C{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}^{q_{1}}} \|f\|^{q_{1}}_{M\dot{K}^{\alpha, \lambda}_{q_{1}, p_{1}(\cdot)}(\mathbb{R}^{n})}\sup\limits_{L\in\mathbb{Z}}2^{-L\lambda q_{1}}{\sum\limits_{k = -\infty}^{L}} 2^{\lambda kq_{1}}{\sum\limits_{j = L+1}^{\infty}}2^{(j-k)(d_{2}+\lambda)\frac{q_{1}}{2}}\\ \leq& C{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}^{q_{1}}} \|f\|^{q_{1}}_{M\dot{K}^{\alpha, \lambda}_{q_{1}, p_{1}(\cdot)}(\mathbb{R}^{n})}. \end{align*} $

情形2 若$ 0<q_{1}\leq1 $, 由已知条件$ d_{2}+\lambda<0 $, 运用$ (2.1) $式, 用$ q_{1} $代替$ q_{1}/q_{2} $可得

$ \begin{align*} &\rm D_{3}\\ \leq& C{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}^{q_{1}}}\sup\limits_{L\in\mathbb{Z}}2^{-L\lambda q_{1}}{\sum\limits_{k = -\infty}^{L}}{\sum\limits_{j = k+3}^{L}}\|f_{j}\|^{q_{1}}_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}2^{\alpha jq_{1}} 2^{d_{2}(j-k)q_{1}}\\ &+C{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}^{q_{1}}}\sup\limits_{L\in\mathbb{Z}}2^{-L\lambda q_{1}}{\sum\limits_{k = -\infty}^{L}}{\sum\limits_{j = L+1}^{\infty}}\|f_{j}\|^{q_{1}}_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}2^{\alpha jq_{1}} 2^{d_{2}(j-k)q_{1}}\\ \leq& C{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}^{q_{1}}} \|f\|^{q_{1}}_{M\dot{K}^{\alpha, \lambda}_{q_{1}, p_{1}(\cdot)}(\mathbb{R}^{n})}\\ &+C{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}^{q_{1}}}\sup\limits_{L\in\mathbb{Z}}2^{-L\lambda q_{1}}{\sum\limits_{k = -\infty}^{L}}{\sum\limits_{j = L+1}^{\infty}}2^{\lambda jq_{1}}2^{d_{2}(j-k)q_{1}}2^{-\lambda jq_{1}}\sum\limits_{m = -\infty}^{j}2^{\alpha mq_{1}}\|f_{m}\|^{q_{1}}_{L^{p_{1}(\cdot)}(\mathbb{R}^{n})}\\ \leq& C{\sum\limits_{|\gamma| = m-1}}{\|D^{\gamma}A\|_{\dot{\bigwedge}_{\beta}(\mathbb{R}^{n})}^{q_{1}}} \|f\|^{q_{1}}_{M\dot{K}^{\alpha, \lambda}_{q_{1}, p_{1}(\cdot)}(\mathbb{R}^{n})}. \end{align*} $

这样就完成了定理1.3的证明.定理1.4的证明与上述证明类似.