记$ S^{n-1}$为$ {\Bbb R}^n(n\geq2)$中的单位球面,其上的Lebesgue测度用$ {\rm d}\sigma={\rm d}\sigma(x') $表示.设函数$ \Omega (x, z)$满足

(1.1) $\Omega (x, \lambda z)=\Omega (x, z), \ \ \forall x, z\in {\Bbb R}^{n}, \forall \lambda>0$

与消失条件

(1.2) $\int_{S^{n-1}}\Omega(x, z'){\rm d}(z')=0, \ \ \forall x\in {\Bbb R}^{n}, \;\;\; z'=\frac{z}{\mid z\mid}, \ \ \forall z\in {\Bbb R}^{n}\setminus\{0\}.$

变量核Marcinkiewicz积分$\mu_{\Omega}$定义为

(1.3) $\mu_{\Omega}(f)(x)=\left(\int_{0}^{\infty}|F_{\Omega, t}(x)|^{2}\frac{{\rm d}t}{t^{3}}\right)^{1/2}, $

其中

$F_{\Omega, t}(x)=\int_{|x-y|\leq t}\frac{\Omega(x, x-y)}{|x-y|^{n-1}}f(y){\rm d}y.$

设$b\in {\rm BMO}({\Bbb R}^n)$,变量核Marcinkiewicz积分交换子$\mu^{b}_{\Omega}(f)$定义为

(1.4) $\mu^{b}_{\Omega}(f)(x)=\left(\int_{0}^{\infty}\bigg|\int_{|x-y|\leq t}\frac{\Omega(x, x-y)}{|x-y|^{n-1}}[b(x)-b(y)]f(y){\rm d}y\bigg|^{2}\frac{{\rm d}t}{t^{3}}\right)^{1/2}. $

Stein^[1]首先证明了当$\Omega(z)=\Omega(x, z)$满足$\Omega\in{\rm Lip}_{\alpha}(S^{n-1})(0<\alpha\leq1)$时, Marcinkiewicz积分$\mu_{\Omega}$是$(p, p)$型$(1<p\leq2)$和弱$(1, 1)$型的.这里$\Omega\in {\rm Lip}_{\alpha}(S^{n-1})(0<\alpha\leq1)$是指,存在一个常数$C$,使得

(1.5) $|\Omega(x, y')-\Omega(x, z')|\leq C|y'-z'|^{\alpha}, \;\;\; \forall y', z'\in S^{n-1}.$

Calderón等^[2]考虑了带变量核的奇异积分算子$T_{\Omega}$的$L^{p}$有界性.他们发现这类算子与变系数的二阶线性椭圆方程密切相关.近年来,关于变量核奇异积分算子的有界性受到了人们的广泛关注,例如, Ding等^[3]得到了带变量核的Marcinkiewicz积分算子的$L^{p}$有界性; Tao等^[4]证明了带变量核的Marcinkiewicz积分算子在齐次Morrey-Herz空间上的有界性;有关变量核Marcinkiewicz积分的相关结果见文献[5-6].

2006年, Cruz-Uribe等^[7]证明了只要Hardy-Littlewood极大算子在变指标$L^{q(\cdot)}$上有界,调和分析中的很多经典算子,如极大算子、奇异积分算子等等都是有界的. 2012年, Wang, Fu和Liu^[8]得到了高阶Marcinkiewicz积分交换子在变指标Lebesgue空间上的有界性.Liu和Wang^[9]考虑了Marcinkiewicz积分$\mu_{\Omega}$及其交换子在变指标Herz空间上的有界性.Tao等^[10]证明了Marcinkiewicz积分$\mu_{\Omega}$及其交换子在变指标Morrey空间上的有界性,最近, Wang和Xu^[11]又得到了多线性Calderón-Zygmund算子及其与BMO函数生成的交换子在变指标Morrey空间上的有界性,受以上研究的启发,本文的主要目的是研究带变量核的Marcinkiewicz积分$\mu_{\Omega}$以及由$\mu_{\Omega}$与BMO函数$b$生成的交换子$\mu^{b}_{\Omega}$在变指标Morrey空间上的有界性.

首先给出一些定义与记号.

设$ k\in Z$,记$ B_k=B(0, 2^k)=\{x\in {\Bbb R}^n:|x|\leq 2^k\}$,及$ C_k=B_k\setminus B_{k-1}$,并记$\chi_k=\chi_{C_k}$为集$ C_k$的特征函数.

定义1.1 给定一个可测函数$q(\cdot):{\Bbb R}^n\rightarrow[1, \infty)$,设$f$是可测函数,对某个$\eta>0$,变指标Lebesgue空间$L^{q(\cdot)}({\Bbb R}^n)$定义如下

$L^{q(\cdot)}({\Bbb R}^n)=\bigg\{f:\int_{{\Bbb R}^n}(\frac{|f(x)|}{\eta})^{q(\cdot)}{\rm d}x<\infty, \eta>0\bigg\}, $

其范数为

$\|f\|_{L^{q(\cdot)}({\Bbb R}^n)}={\rm inf}\bigg\{\eta>0:\int_{{\Bbb R}^n}(\frac{|f(x)|}{\eta})^{q(\cdot)}{\rm d}x\leq1\bigg\}.$

易见,如果$q(\cdot)=q_{0}$是常数,那么$L^{q(\cdot)}({\Bbb R}^n)$成为标准的Lebesgue空间$L^{q_{0}}({\Bbb R}^n)$.记

${\cal P}({\Bbb R}^n)=\{q(\cdot):1<q_{-}, q^{+}<\infty\}.$

用${\cal B}({\Bbb R}^n)$表示使得Hardy-Littlewood极大算子在$L^{q(\cdot)}({\Bbb R}^n)$上有界的所有${\cal P}({\Bbb R}^n)$中的函数$q(\cdot)$构成的集合^[7].

定义1.2^[12] 设$q\in L^{\infty}$并且$1<q(\cdot)\leq\infty, $若存在一个常数$C$,使得对任意的$x\in{\Bbb R}^n$和$r>0$,可测函数$u(x, r):{\Bbb R}^n\times(0, \infty)\rightarrow(0, \infty)$满足

(1.6) $\mathop \sum \limits_{j = 0}^\infty \frac{\|\chi_{B(x, r)}\|_{L^{q(\cdot)}({\Bbb R}^n)}}{\|\chi_{B(x, 2^{j+1}r)}\|_{L^{q(\cdot)}({\Bbb R}^n)}}u(x, 2^{j+1}r)<C u(x, r), $

则称$u(x, r)$是$L^{q(\cdot)}({\Bbb R}^n)$上的一个Morrey权函数.用${\Bbb W}_{q(\cdot)}$来表示Morrey权函数类.

定义1.3^[12] 设$q(\cdot)\in {\cal P}({\Bbb R}^n), u(x, r)\in{\Bbb W}_{q(\cdot)}$,变指标Morrey空间定义为

${\cal M}_{q(\cdot), u}({\Bbb R}^n)=\bigg\{f:\|f\|_{{\cal M}_{q(\cdot), u}}=\sup\limits_{z\in{\Bbb R}^n, r>0}\frac{1}{u(z, r)}\|\chi_{B(z, r)}f\|_{L^{q(\cdot)}({\Bbb R}^n)}<\infty\bigg\}.$

容易看出,当$\lambda, q$为常数,且$u(x, r)=r^{\lambda}, 0\leq\lambda<n, q(\cdot)=q$时,变指标的Morrey空间${\cal M}_{q(\cdot), u}({\Bbb R}^n)$就是经典的Morrey空间$M_{q, \lambda}({\Bbb R}^n).$

BMO空间定义如下

${\rm BMO}({\Bbb R}^n)=\bigg\{b\in L^{1}_{{\rm loc}}({\Bbb R}^n):\|b\|_{{\rm BMO}({\Bbb R}^n)}:=\sup\limits_{Q\subset{\Bbb R}^n}\frac{1}{|Q|}\int_{Q}|b(x)-b_{Q}|{\rm d}x<\infty\bigg\}, $

其中$b_{Q}=\frac{1}{|Q|}\int_{Q}f(y){\rm d}y.$

$\dot{\Lambda}_{\beta}({\Bbb R}^n)=\bigg\{f\in L^{1}_{{\rm loc}}({\Bbb R}^n):\|f\|_{\dot{\Lambda}_{\beta}({\Bbb R}^n)}:=\sup\limits_{x, y\in{\Bbb R}^n, x\neq y}\frac{|f(x)-f(y)|}{|x-y|^{\beta}}<\infty\bigg\}.$

给定$f\in L^{1}_{{\rm loc}}({\Bbb R}^n)$, Hardy-Littlewood极大算子定义为

$Mf(x)=\sup\limits_{Q\ni x}\frac{1}{|Q|}\int_{Q}|f(y)|{\rm d}y, $

这里上确界取遍所有边平行于坐标轴的方体$Q\subset{\Bbb R}^n$, $|Q|$表示$Q$的Lebesgue测度.

给定$\lambda\in(0, 1)$及${\Bbb R}^n$上可测函数$f$,局部Sharp极大算子$M_{\lambda}^{\sharp}$定义为

$M_{\lambda}^{\sharp}f(x)=\sup\limits_{Q\ni x}\inf\limits_{c\in{\Bbb R}}((f-c)\chi_{Q})^{\ast}(\lambda|Q|).$

本文主要结果如下

定理1.4 设变量核Marcinkiewicz积分$\mu_{\Omega}$由(1.3)式所定义, $\Omega(x, z)$满足条件(1.1), (1.2)和(1.5),且$q(\cdot)\in {\cal B}({\Bbb R}^n)$,则$\mu_{\Omega}$在变指标Morrey空间上是有界的,即存在与$f$无关的常数$C>0$,使得对任意的$f\in{\cal M}_{q(\cdot), \mu}$,有

$\|\mu_{\Omega}(f)\|_{{\cal M}_{q(\cdot), \mu}}\leq C\|f\|_{{\cal M}_{q(\cdot), \mu}}.$

定理1.5 设$b\in {\rm BMO}({\Bbb R}^n)$, $\Omega(x, z)$满足条件(1.1), (1.2)和(1.5), $q(\cdot)\in {\cal B}({\Bbb R}^n)$,且函数$u$满足

(1.7) $\mathop \sum \limits_{j = 1}^\infty (j+1)\frac{\|\chi_{B(x, r)}\|_{L^{q(\cdot)}({\Bbb R}^n)}}{\|\chi_{B(x, 2^{j+1}r)}\|_{L^{q(\cdot)}({\Bbb R}^n)}}\frac{u(x, 2^{j+1}r)}{u(x, r)}<C.$

则$\mu_{\Omega}$在变指标Morrey空间上是有界的,即存在与$f$无关的常数$C>0$,使得对任意的$f\in{\cal M}_{q(\cdot), \mu}$,有

$\|\mu_{\Omega}(f)\|_{{\cal M}_{q(\cdot), \mu}}\leq C\|f\|_{{\cal M}_{q(\cdot), \mu}}.$

证明定理,需要以下引理

引理2.1^[13] (广义Hölder不等式) 设$q(\cdot)\in{\cal B}({\Bbb R}^n)$且$f\in L^{q(\cdot)}({\Bbb R}^n), g\in L^{q'(\cdot)}({\Bbb R}^n)$,则$f, g$在${\Bbb R}^n$上可积,且有

$\int_{{\Bbb R}^{n}}|f(x)g(y)|{\rm d}x \leq r_{q}\|f\|_{L^{q(\cdot)}({\Bbb R}^n)}\|g\|_{L^{q'(\cdot)}({\Bbb R}^n)}, $

其中$r_{q}=1+\frac{1}{q_{-}}-\frac{1}{q_{+}}$.

引理2.2^[14] 设$q(\cdot)\in {\cal B}({\Bbb R}^n)$,则存在常数$C>0$,使得对${\Bbb R}^n$中所有的球$B$,有

(2.1) $C^{-1}\leq\frac{1}{|B|}\|\chi_{B}\|_{L^{q(\cdot)}({\Bbb R}^n)}\|\chi_{B}\|_{L^{q'(\cdot)}({\Bbb R}^n)}\leq C .$

引理2.3^[15] 设$g\in L^{1}_{{\rm loc}}({\Bbb R}^n)$,可测函数$\varphi$满足对任意$\alpha>0$有

$|\{x:|\varphi(x)|>\alpha\}|<\infty, $

则

(2.2) $\int_{{\Bbb R}^{n}}|\varphi(x)g(x)|{\rm d}x \leq C_n\int_{{\Bbb R}^{n}}M^{\sharp}_{\lambda_n}\varphi(x)g(x), $

其中$\lambda_n\in(0, 1).$

引理2.4 若$\Omega(x, z)$满足条件(1.1), (1.2)和(1.5).假设$q(\cdot)\in{\cal B}({\Bbb R}^n)$,则存在常数$C$,使得对任意的$f\in L^{q(\cdot)}({\Bbb R}^n)$,有

$\|\mu_{\Omega}(f)\|_{L^{q(\cdot)}({\Bbb R}^n)}\leq C_p\|f\|_{L^{q(\cdot)}({\Bbb R}^n)}.$

证 令$f\in L^{\infty}_{c}.$对于任意$g\in L^{q'(\cdot)}({\Bbb R}^n)\subset L^{1}_{{\rm loc}}({\Bbb R}^n), $由引理2.3,我们有

$ \begin{eqnarray*} \int_{{\Bbb R}^{n}}|\mu_{\Omega}(f)(x)g(x)|{\rm d}x &\leq&c_n\int_{{\Bbb R}^{n}}M^{\sharp}_{\lambda_n}(\mu_{\Omega}(f))(x)Mg(x){\rm d}x\\ &\leq &c_n\int_{{\Bbb R}^{n}}(1/\lambda_n)^{1/\delta}(\mu_{\Omega}(f))^{\sharp}_{\delta}(x)Mg(x){\rm d}x\\ &\leq &c_n\int_{{\Bbb R}^{n}}Mf(x)Mg(x){\rm d}x\\ &\leq &c_nr_{p}\|Mf\|_{L^{q(\cdot)}({\Bbb R}^n)}\|Mg\|_{L^{q'(\cdot)}({\Bbb R}^n)}\\ &\leq &C_{p}\|f\|_{L^{q(\cdot)}({\Bbb R}^n)}\|g\|_{L^{q'(\cdot)}({\Bbb R}^n)}, \end{eqnarray*}$

其中$C_{p}=c_nr_{p}\|M\|_{L^{q(\cdot)}({\Bbb R}^n)\rightarrow L^{q(\cdot)}({\Bbb R}^n)}\|M\|_{L^{q'(\cdot)}({\Bbb R}^n)\rightarrow L^{q'(\cdot)}({\Bbb R}^n)}.$从而

$\|\mu_{\Omega}(f)\|_{L^{q(\cdot)}({\Bbb R}^n)}\leq C_{p}\|f\|_{L^{q(\cdot)}({\Bbb R}^n)}.$

由Hardy-Littlewood极大算子$M(f)$与局部Sharp极大算子$M^{\sharp}_{\lambda}(f)$的性质,引理2.4得证.

引理2.5^[14] 设$b\in {\rm BMO}, m\in{\Bbb N}, i, j\in{\Bbb Z}$,满足$i<j, $则有

(2.3) $ C^{-1}\|b\|^{m}_{{\rm BMO}({\Bbb R}^n)}\leq\sup\limits_{B}\frac{\|(b-b_{B})^{m}\chi_{B}\|_{L^{q(\cdot)}({\Bbb R}^n)}}{\|\chi_{B}\|_{L^{q(\cdot)}({\Bbb R}^n)}}\leq\|b\|^{m}_{{\rm BMO}({\Bbb R}^n)}, $

(2.4) $ \|(b-b_{B_i})^{m}\chi_{B_j}\|_{L^{q(\cdot)}({\Bbb R}^n)}\leq C(j-i)^{m}\|b\|^{m}_{{\rm BMO}({\Bbb R}^n)}\|\chi_{B}\|_{L^{q(\cdot)}({\Bbb R}^n)}, $

其中$B=(x, r), B_i=(x, 2^{ir})$.

引理2.6 若$\Omega(x, z)$满足条件(1.1), (1.2)和(1.5).假设$q(\cdot)\in {\cal B}({\Bbb R}^n)$,则存在常数$C$,使得对任意的$f\in L^{q(\cdot)}({\Bbb R}^n)$,有

$\|\mu^{b}_{\Omega}(f)\|_{L^{q(\cdot)}({\Bbb R}^n)}\leq C\|b\|_{{\rm BMO}({\Bbb R}^n)}\|f\|_{L^{q(\cdot)}({\Bbb R}^n)}.$

利用文献[8]与[16]中相应定理的证明方法可证引理2.6.

定理1.4的证明 设$f\in{\cal M}_{q(\cdot), u}.$对任意的$z\in{\Bbb R}^n$和$r>0$,分解$f(x)=f_{1}(x)+f_{2}(x)$,其中$f_{1}=f\chi_{B(z, 2r)}, f_{2}=\sum\limits_{j=1}^{\infty}f_{j}, f_{j}=f\chi_{B(z, 2^{j+1}r)\setminus B(z, 2^{j}r)}, $则

$ \begin{eqnarray*} \frac{1}{u(z, r)}\|\chi_{B(z, r)}\mu_\Omega(f)\|_{L^{q(\cdot)}({\Bbb R}^n)} &\leq&\frac{1}{u(z, r)}\|\chi_{B(z, r)}\mu_\Omega(f_1)\|_{L^{q(\cdot)}({\Bbb R}^n)}\\ &&+\frac{1}{u(z, r)}\|\chi_{B(z, r)}\mu_\Omega(f_2)\|_{L^{q(\cdot)}({\Bbb R}^n)}\\ &=:&E_1+E_2. \end{eqnarray*}$

先估计$E_1$.由引理2.4, $\mu_{\Omega}(f)$在$L^{q(\cdot)}({\Bbb R}^n)$上有界,所以我们有

$ \begin{eqnarray*} E_1 &\leq&C\frac{1}{u(z, 2r)}\|\chi_{B(z, 2r)}f\|_{L^{q(\cdot)}({\Bbb R}^n)}\\ &\leq&C\sup\limits_{y\in{\Bbb R}^n, r>0}\frac{1}{u(y, r)}\|\chi_{B(y, r)}f\|_{L^{q(\cdot)}({\Bbb R}^n)}\\ &\leq&C\|f\|_{{\cal M}_{q(\cdot), u}}. \end{eqnarray*} $

下面估计$E_2$.注意到当$x\in B(z, r), y\in B(z, 2^{j+1}r)\setminus B(z, 2^{j}r)$时,有

$2^{j-1}r\leq|y-z|-|x-z|\leq|x-y|<t.$

因此,应用Hölder不等式有

$\begin{eqnarray*} |\mu_\Omega(f_2)(x)| &\leq&\sum\limits_{j=1}^{\infty}\bigg(\int_{B(z, 2^{j+1}r)\setminus B(z, 2^{j}r)}\frac{\Omega(x, x-y)}{|x-y|^{n-1}}|f(y)|{\rm d}y\bigg)\bigg(\int^{\infty}_{2^{(j-1)n}r^{n}}\frac{{\rm d}t}{t^{3}}\bigg)^{1/2}\\ &\leq&C\sum\limits_{j=1}^{\infty}\frac{1}{|2^{(j+1)n}r^{n}|^{\frac{1}{n}}}\int_{B(z, 2^{j+1}r)\setminus B(z, 2^{j}r)}\frac{\Omega(x, x-y)}{|x-y|^{n-1}}|f(y)|{\rm d}y. \end{eqnarray*}$

又由条件(1.5)知$\Omega$有界,利用引理2.1,有

$\begin{eqnarray*} |\mu_\Omega(f_2)(x)|&\leq&C\sum\limits_{j=1}^{\infty}\frac{1}{|2^{(j+1)n}r^{n}|^{\frac{1}{n}}}\int_{B(z, 2^{j+1}r)\setminus B(z, 2^{j}r)}\frac{|f(y)|}{|x-y|^{n-1}}{\rm d}y\\ &\leq&C\sum\limits_{j=1}^{\infty}\frac{1}{|2^{(j+1)n}r^{n}|^{\frac{1}{n}}}\frac{1}{|2^{(j+1)n}r^{n}|^{\frac{n-1}{n}}}\int_{B(z, 2^{j+1}r)}|f(y)|{\rm d}y\\ &\leq&C\sum\limits_{j=1}^{\infty}\frac{1}{2^{(j+1)n}r^{n}}\int_{B(z, 2^{j+1}r)}|f(y)|{\rm d}y\\ &\leq&C\sum\limits_{j=1}^{\infty}\frac{1}{2^{(j+1)n}r^{n}}\|f\chi_{B(z, 2^{j+1}r)}\|_{L^{q(\cdot)}({\Bbb R}^n)}\|\chi_{B(z, 2^{j+1}r)}\|_{L^{q'(\cdot)}({\Bbb R}^n)}.\end{eqnarray*}$

因此

$|\chi_{B(z, r)}\mu_\Omega(f_2)(x)|\leq C\mathop \sum \limits_{j = 1}^\infty \frac{1}{2^{(j+1)n}r^{n}}\|f\chi_{B(z, 2^{j+1}r)}\|_{L^{q(\cdot)}({\Bbb R}^n)}\|\chi_{B(z, 2^{j+1}r)}\|_{L^{q'(\cdot)}({\Bbb R}^n)}\cdot\chi_{B(z, r)}.$

上式两端同时取范数$\|\cdot\|_{L^{q(\cdot)}({\Bbb R}^n)}$,可得

$\begin{eqnarray*} \|\chi_{B(z, r)}\mu_\Omega(f_2)(x)\|_{L^{q(\cdot)}({\Bbb R}^n)} &\leq&C\sum\limits_{j=1}^{\infty}\frac{1}{2^{(j+1)n}r^{n}}\|f\chi_{B(z, 2^{j+1}r)}\|_{L^{q(\cdot)}({\Bbb R}^n)}\|\chi_{B(z, 2^{j+1}r)}\|_{L^{q'(\cdot)}({\Bbb R}^n)}\\ &&\cdot\|\chi_{B(z, r)}\|_{L^{q(\cdot)}({\Bbb R}^n)}. \end{eqnarray*} $

在引理2.2中,取$B=B(z, 2^{j+1}r)$,则

(2.5) $\|\chi_{B(z, 2^{j+1}r)}\|_{L^{q'(\cdot)}({\Bbb R}^n)}\leq C\frac{2^{(j+1)n}r^{n}}{\|\chi_{B(z, 2^{j+1}r)}\|_{L^{q(\cdot)}({\Bbb R}^n)}}.$

由(2.5)式知

$\begin{eqnarray*} &&\|\chi_{B(z, r)}\mu_\Omega(f_2)(x)\|_{L^{q(\cdot)}({\Bbb R}^n)} \\&\leq&C\sum\limits_{j=1}^{\infty}\frac{1}{2^{(j+1)n}r^{n}}\|f\chi_{B(z, 2^{j+1}r)}\|_{L^{q(\cdot)}({\Bbb R}^n)}\frac{2^{(j+1)n}r^{n}}{\|\chi_{B(z, 2^{j+1}r)}\|_{L^{q(\cdot)}({\Bbb R}^n)}}\|\chi_{B(z, r)}\|_{L^{q(\cdot)}({\Bbb R}^n)}\\ &\leq&C\sum\limits_{j=1}^{\infty}\frac{\|\chi_{B(z, r)}\|_{L^{q(\cdot)}({\Bbb R}^n)}}{\|\chi_{B(z, 2^{j+1}r)}\|_{L^{q(\cdot)}({\Bbb R}^n)}}\frac{u(z, 2^{(j+1)}r)}{u(z, 2^{(j+1)}r)}\|f\chi_{B(z, 2^{j+1}r)}\|_{L^{q(\cdot)}({\Bbb R}^n)}\\ &\leq&C\sum\limits_{j=1}^{\infty}\frac{\|\chi_{B(z, r)}\|_{L^{q(\cdot)}({\Bbb R}^n)}u(z, 2^{(j+1)}r)}{\|\chi_{B(z, 2^{j+1}r)}\|_{L^{q(\cdot)}({\Bbb R}^n)}}\sup\limits_{z\in{\Bbb R}^n, r>0}\frac{1}{u(z, r)}\|\chi_{B(z, r)}f\|_{L^{q(\cdot)}({\Bbb R}^n)}. \end{eqnarray*}$

由定义1.2,若$u\in{\Bbb W}_{q(\cdot)}$,有

$\mathop \sum \limits_{j = 1}^\infty \frac{\|\chi_{B(z, r)}\|_{L^{q(\cdot)}({\Bbb R}^n)}}{\|\chi_{B(z, 2^{j+1}r)}\|_{L^{q(\cdot)}({\Bbb R}^n)}}\frac{u(z, 2^{(j+1)}r)}{u(z, r)}<C.$

因此

$E_2\leq C\|f\|_{{\cal M}_{q(\cdot), u}}.$

从而定理1.4得证.

定理1.5的证明 设$f\in{\cal M}_{q(\cdot), u}, b\in {\rm BMO}({\Bbb R}^n)$对任意的$z\in{\Bbb R}^n$和$r>0$,分解$f(x)=f_{1}(x)+f_{2}(x)$,其中$f_{1}=f\chi_{B(z, 2r)}, f_{2}=\sum\limits_{j=1}^{\infty}f_{j}, f_{j}=f\chi_{B(z, 2^{j+1}r)\setminus B(z, 2^{j}r)}, $记

$ \begin{eqnarray*} &&\frac{1}{u(z, r)}\|\chi_{B(z, r)}\mu_{\Omega}^{b}(f)\|_{L^{q(\cdot)}({\Bbb R}^n)}\\ &\leq&\frac{1}{u(z, r)}\|\chi_{B(z, r)}\mu_{\Omega}^{b}(f_1)\|_{L^{q(\cdot)}({\Bbb R}^n)}+\frac{1}{u(z, r)}\|\chi_{B(z, r)}\mu_{\Omega}^{b}(f_2)\|_{L^{q(\cdot)}({\Bbb R}^n)}\\ &=:&F_1+F_2. \end{eqnarray*}$

由引理2.6及定理1.4中关于$E_1$的估计,我们有$F_1\leq C\|f\|_{{\cal M}_{q(\cdot), u}}$.

以下估计$F_2, $当$x\in B(z, r), y\in B(z, 2^{j+1}r)\setminus B(z, 2^{j}r)$时,有

$2^{j-1}r\leq|y-z|-|x-z|\leq|x-y|<t.$

因此,应用Hölder不等式有

$\begin{array}{l} \|\mu_\Omega^{b}(f)(x)\| =\bigg(\int^{\infty}_{0}\bigg|\int_{(B(z,r))^{c}\bigcap\{y:|x-y|<t\}}\frac{\Omega(x,x-y)}{|x-y|^{n-1}}[b(x)-b(y)]f(y)\bigg|^{2}\frac{{\rm d}t}{t^{3}}\bigg)^{\frac{1}{2}}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\leq\sum\limits_{j=1}^{\infty}\bigg(\int_{B(z,2^{j+1}r)\setminus B(z,2^{j}r)}\frac{\Omega(x,x-y)}{|x-y|^{n-1}}[b(x)-b(y)]|f(y)|{\rm d}y\bigg)\bigg(\int^{\infty}_{2^{(j-1)n}r^{n}}\frac{{\rm d}t}{t^{3}}\bigg)^{\frac{1}{2}}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\leq C\sum\limits_{j=1}^{\infty}\frac{1}{|2^{(j+1)n}r^{n}|^{\frac{1}{n}}}\cdot\int_{B(z,2^{j+1}r)\setminus B(z,2^{j}r)}\frac{\Omega(x,x-y)}{|x-y|^{n-1}}[b(x)-b(y)]|f(y)|{\rm d}y. \end{array}$

应用广义Hölder不等式,有

$ \begin{eqnarray*} \|\mu_\Omega^{b}(f)(x)\| &\leq&C\sum\limits_{j=1}^{\infty}\frac{1}{|2^{(j+1)}r|^{\frac{1}{n}}}\cdot\int_{B(z, 2^{j+1}r)\setminus B(z, 2^{j}r)}\frac{|b(x)-b(y)|}{|x-y|^{n-1}}|f(y)|{\rm d}y\\ &\leq&C\sum\limits_{j=1}^{\infty}\frac{1}{|2^{(j+1)n}r^{n}|^{\frac{1}{n}}}\frac{1}{|2^{(j+1)}r|^{\frac{n-1}{n}}}\int_{B(z, 2^{j+1}r)}|b(x)-b(y)||f(y)|{\rm d}y\\ &\leq&C\sum\limits_{j=1}^{\infty}\frac{1}{2^{(j+1)n}r^{n}}\int_{B(z, 2^{j+1}r)}|b(x)-b(y)||f(y)|{\rm d}y\\ &\leq&C\sum\limits_{j=1}^{\infty}\frac{1}{2^{(j+1)n}r^{n}}|b(x)-b_{B(z, r)}|\int_{B(z, 2^{j+1}r)}|f(y)|{\rm d}y\\ &&+ C\sum\limits_{j=1}^{\infty}\frac{1}{2^{(j+1)n}r^{n}}\int_{B(z, 2^{j+1}r)}|b_{B(z, r)}-b(y)||f(y)|{\rm d}y\\ &\leq&C\sum\limits_{j=1}^{\infty}\frac{1}{2^{(j+1)n}r^{n}}|b(x)-b_{B(z, r)}|\|f\chi_{B(z, 2^{(j+1)}r)}\|_{L^{q(\cdot)}({\Bbb R}^n)}\|\chi_{B(z, 2^{(j+1)}r)}\|_{L^{q'(\cdot)}({\Bbb R}^n)}\\ &&+C\sum\limits_{j=1}^{\infty}\frac{1}{2^{(j+1)n}r^{n}}\|f\chi_{B(z, 2^{(j+1)}r)}\|_{L^{q(\cdot)}({\Bbb R}^n)}\|(b_{B(z, r)}-b)\chi_{B(z, 2^{(j+1)}r)}\|_{L^{q'(\cdot)}({\Bbb R}^n)}, \end{eqnarray*}$

现记$I=\|\chi_{B(z, r)}\mu_{\Omega}^{b}(f_2)\|_{L^{q(\cdot)}({\Bbb R}^n)}$,则

$ \begin{eqnarray*} I &\leq&C\sum\limits_{j=1}^{\infty}\frac{1}{2^{(j+1)n}r^{n}}\|f\chi_{B(z, 2^{(j+1)}r)}\|_{L^{q(\cdot)}({\Bbb R}^n)}\|\chi_{B(z, 2^{(j+1)}r)}\|_{L^{q'(\cdot)}({\Bbb R}^n)}\\ &&\cdot\|(b-b_{B(z, r)})\chi_{B(z, r)}\|_{L^{q(\cdot)}({\Bbb R}^n)}\\ &&+C\sum\limits_{j=1}^{\infty}\frac{1}{2^{(j+1)n}r^{n}}\|f\chi_{B(z, 2^{(j+1)}r)}\|_{L^{q(\cdot)}({\Bbb R}^n)}\|(b_{B(z, r)}-b)\chi_{B(z, 2^{(j+1)}r)}\|_{L^{q'(\cdot)}({\Bbb R}^n)}\\ &&\cdot\|\chi_{B(z, r)}\|_{L^{q(\cdot)}({\Bbb R}^n)}. \end{eqnarray*} $

由引理2.5,有

$\begin{eqnarray*} I &\leq&C\sum\limits_{j=1}^{\infty}\frac{1}{2^{(j+1)n}r^{n}}\|f\chi_{B(z, 2^{(j+1)}r)}\|_{L^{q(\cdot)}({\Bbb R}^n)}\|\chi_{B(z, 2^{(j+1)}r)}\|_{L^{q'(\cdot)}({\Bbb R}^n)}\|b\|_{{\rm BMO}} \|\chi_{B(z, r)}\|_{L^{q(\cdot)}({\Bbb R}^n)}\\ &&+C\sum\limits_{j=1}^{\infty}\frac{j+1}{2^{(j+1)n}r^{n}}\|f\chi_{B(z, 2^{(j+1)}r)}\|_{L^{q(\cdot)}({\Bbb R}^n)}\|\chi_{B(z, 2^{(j+1)}r)}\|_{L^{q'(\cdot)}({\Bbb R}^n)}\|b\|_{{\rm BMO}} \|\chi_{B(z, r)}\|_{L^{q(\cdot)}({\Bbb R}^n)}\\ &\leq&C\sum\limits_{j=1}^{\infty}\frac{j+1}{2^{(j+1)n}r^{n}}\|b\|_{{\rm BMO}}\|f\chi_{B(z, 2^{(j+1)}r)}\|_{L^{q(\cdot)}({\Bbb R}^n)}\|\chi_{B(z, 2^{(j+1)}r)}\|_{L^{q'(\cdot)}({\Bbb R}^n)} \|\chi_{B(z, r)}\|_{L^{q(\cdot)}({\Bbb R}^n)}. \end{eqnarray*} $

将(2.5)式代入上式,可得

$ \begin{eqnarray*} I &\leq&C\sum\limits_{j=1}^{\infty}\frac{j+1}{2^{(j+1)n}r^{n}}\|b\|_{{\rm BMO}({\Bbb R}^n)}\|f\chi_{B(z, 2^{(j+1)}r)}\|_{L^{q(\cdot)}({\Bbb R}^n)}\\ &&\cdot\|b\|_{{\rm BMO}({\Bbb R}^n)} \frac{2^{(j+1)n}r^{n}}{\|\chi_{B(z, 2^{(j+1)}r)}\|_{L^{q(\cdot)}({\Bbb R}^n)}} \|\chi_{B(z, r)}\|_{L^{q(\cdot)}({\Bbb R}^n)}\\ &\leq&C\sum\limits_{j=1}^{\infty}(j+1)\|b\|_{{\rm BMO}({\Bbb R}^n)}\frac{\|\chi_{B(z, r)}\|_{L^{q(\cdot)}({\Bbb R}^n)}}{\|\chi_{B(z, 2^{(j+1)}r)}\|_{L^{q(\cdot)} ({\Bbb R}^n)}}\frac{u(z, 2^{(j+1)}r)}{u(z, 2^{(j+1)}r)}\|f\chi_{B(z, 2^{(j+1)}r)}\|_{L^{q(\cdot)}({\Bbb R}^n)}\\ &\leq&C\sum\limits_{j=1}^{\infty}(j+1)\|b\|_{{\rm BMO}({\Bbb R}^n)}\frac{\|\chi_{B(z, r)}\|_{L^{q(\cdot)}({\Bbb R}^n)}u(z, 2^{(j+1)}r)}{\|\chi_{B(z, 2^{(j+1)}r)}\|_{L^{q(\cdot)} ({\Bbb R}^n)}}\\ &&\cdot\sup\limits_{y\in{\Bbb R}^n, r>0}\frac{1}{u(y, r)}\|\chi_{B(y, r)}f\|_{L^{q(\cdot)}({\Bbb R}^n)}. \end{eqnarray*}$

因此,由(1.7)式,可得

$F_2\leq C\|b\|_{{\rm BMO}({\Bbb R}^n)}\|f\|_{{\cal M}_{q(\cdot), u}}.$

至此,定理1.5得证.