1990年, Torchinsky和Wang^[1]证明了$\mu_{\Omega}$的加权有界性.随后, Chen等^[2]得到了当$\Omega$在$S^{n-1}$上满足一类$L^{r}-{\hbox{Dini}}(r\geq1)$条件时, 对某些$p\leq1$, 带变量核的奇异积分算子是$H^{p}$到$L^{p}$有界的.随后, Ding等^{[3, 4]}得到了带变量核的Marcinkiewicz算子的$L^{p}$有界性. 2005年, 陈冬香等^[5]讨论了带粗糙核的Marcinkiewicz算子在Herz空间上的有界性. 2010年, 陶双平等^[6]讨论了带变量核的Marcinkiewicz算子在齐次Morrey-Herz空间上的有界性. 2010年, 肖强等^[7]讨论了带粗糙核的Marcinkiewicz算子在加权Morrey-Herz空间上的有界性.受此启发, 本文讨论了带变量核的Marcinkiewicz算子在加权Herz空间上的有界性.为此先介绍相关的定义.

记$S^{n-1}$为${\mathbb{R}}^{n}(n\geq2)$中的单位球面, $d\sigma$表示单位球面上的Lebesgue测度, 称定义在${\mathbb{R}}^{n}\times{\mathbb{R}}^{n}$上的函数$\Omega(x, z)\in{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}(r\geq1)$是指

$ \big\|\Omega\big\|_{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}d\sigma(z')\right)^{\frac{1}{r}}<\infty, $

(1)

其中$z'=\dfrac{z}{|z|}, z\in{\mathbb{R}}^{n}\backslash\{0\}, \Omega$是零阶齐次的是指

$ \Omega(x, \lambda~z)=\Omega(x, z), \forall x, z\in{\mathbb{R}}^{n}, \lambda>0. $

(2)

设$\Omega$满足消失矩条件:

$ \int_{S^{n-1}}\Omega(x, z')d(z')=0, \forall~x\in{\mathbb{R}}^{n}. $

(3)

设$b\in BMO({\mathbb{R}}^{n})$, 带变量核的Marcinkiewicz积分算子与$b$生成的交换子$\mu_{\Omega}^{b}$定义为

$ \mu_{\Omega}^{b}(f)(x)=\left\{\int_{0}^{\infty}\big|\int_{|x-y|\leq~t}\dfrac{\Omega(x, x-y)}{|x-y|^{n-1}}(b(x)-b(y))f(y)dy\big|^{2}\dfrac{dt}{t^{3}}\right\}^{\frac{1}{2}}. $

加权Herz空间定义如下:

定义^[8]  设$0<\alpha<\infty, 0<p<\infty, 1\leq q<\infty, $且$\omega_{1}, \omega_{2}$是非负权函数, 加权Herz空间$K^{\alpha, p}_{q}(\omega_{1}, \omega_{2})$的定义为

$ K^{\alpha, p}_{q}(\omega_{1}, \omega_{2})=\{f\in~L^{q}_{loc}({\mathbb{R}}^{n}\backslash\{0\}, \omega_{2}):\big\|f\big\|_{K^{\alpha, p}_{q}(\omega_{1}, \omega_{2})}<\infty\}, $

其中

$ \big\|f\big\|_{K^{\alpha, p}_{q}(\omega_{1}, \omega_{2})}=\left\{\sum\limits_{k=-\infty}^{\infty}[\omega_{1}(B_{k})]^{\frac{\alpha p}{n}}\big\|f\chi_{k}\big\|^{p}_{L^{q}(\omega_{2})}\right\}^{\frac{1}{p}}, $

这里, $B_{k}=\{x\in{\mathbb{R}}^{n}:|x|\leq2^{k}\}, C_{k}=B_{k}\backslash~B_{k-1}, k\in\mathbb{Z}, \chi_{k}=\chi_{C_{k}}(x)$为集合$C_{k}$的特征函数, 并记$f_{k}=f\chi_{k}.$

本文中$C$在不同的地方表示不同的正常数.

为了证明本文定理, 需要以下引理:

引理 2.1^[9] 如果$\omega\in A_{p}, 1\leq p<\infty, $则存在常数$C>0$及$0<\delta<1, $使得对任意$k, j\in\mathbb{Z}, $有

(ⅰ) $\forall k>j, \dfrac{\omega(B_{k})}{\omega(B_{j})}\leq C\dfrac{|B_{k}|}{|B_{j}|};$

(ⅱ) $\forall k\leq j, \dfrac{\omega(B_{k})}{\omega(B_{j})}\leq C(\dfrac{|B_{k}|}{|B_{j}|})^{\delta};$

(ⅲ) $\forall j, \omega(B_{j})\leq C|B_{j}|essinf\{\omega(y):y\in B_{j}\}.$

引理 2.2^[10] 设核函数$\Omega$满足(1)-(3) 式, $a>0, 1<r\leq\infty, 0<d\leq r$且$-\frac{n}{d}+\frac{(n-1)}{r}<\beta<\infty, $则

$ \left\{\int_{|y|\leq~a|x|}|\Omega(x, x-y)|y|^{\beta}|^{d}dy\right\}^{\frac{1}{d}}\leq C\big|x\big|^{\beta+\frac{n}{d}}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}. $

引理 2.3 ^[10] 设核函数$\Omega$满足(1)-(3) 式, $a>0, 1<r\leq\infty, 0<d\leq r$且$\beta<-\frac{n}{d}.$则

$ \left\{\int_{|y|\geq~a|x|}|\Omega(x, x-y)|y|^{\beta}|^{d}dy\right\}^{\frac{1}{d}}\leq C\big|x\big|^{\beta+\frac{n}{d}}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}. $

引理 2.4 ^[11] 设核函数$\Omega$满足(1)-(3) 式, 对$1\leq q<\infty, $权函数$\omega\in A_{q}, $则存在常数$C>0, $使得$\big\|\mu_{\Omega}^{b}(f)\big\|_{L^{q}(\omega)}\leq C\big\|b\big\|_{{\hbox{BMO}}}\big\|f\big\|_{L^{q}(\omega)}$对任意的$f\in L^{q}(\omega)$成立.

定理  设核函数$\Omega$满足(1)-(3) 式, $0<p<\infty, 1<q<\infty, 0<r\leq\infty, \delta$为引理2.1中的常数且权函数$\omega_{1}, \omega_{2}\in A_{1}, $若$r'<q$且$n(\dfrac{1}{\delta r'}-\dfrac{1}{q})<\alpha<n(\dfrac{1}{r'}-\dfrac{1}{q})+\dfrac{1}{r}, b\in {\hbox{BMO}}, $则带变量核的~Marcinkiewicz~积分算子与函数$b$生成的交换子$\mu_{\Omega}^{b}$在加权Herz空间$K^{\alpha, p}_{q}(\omega_{1}, \omega_{2})$上有界.

证  只需证明存在常数$C>0, $使得$\big\|\mu_{\Omega}^{b}(f)\big\|_{K^{\alpha, p}_{q}(\omega_{1}, \omega_{2})}\leq C\big\|b\big\|_{{\hbox{BMO}}}\big\|f\big\|_{K^{\alpha, p}_{q}(\omega_{1}, \omega_{2})}$对任意的$f\in K^{\alpha, p}_{q}(\omega_{1}, \omega_{2})$成立.

设$f\in K^{\alpha, p}_{q}(\omega_{1}, \omega_{2}), $记$f(x)=\sum\limits_{j=-\infty}^{\infty}f(x)\chi_{j}(x)=\sum\limits_{j=-\infty}^{\infty}f_{j}(x).$于是

$ \begin{aligned} \big\|\mu_{\Omega}^{b}(f)\big\|_{K^{\alpha, p}_{q}(\omega_{1}, \omega_{2})}^{p}=& \sum\limits_{k=-\infty}^{\infty}[\omega_{1}(B_{k})]^{\frac{\alpha p}{n}}\big\|\mu_{\Omega}^{b}(f)\chi_{k}\big\|_{L^{q}(\omega_{2})}^{p}\\ \leq& C\sum\limits_{k=-\infty}^{\infty}[\omega_{1}(B_{k})]^{\frac{\alpha p}{n}}\left\{\sum\limits_{j=-\infty}^{k-2}\big\|\mu_{\Omega}^{b}(f_{j})\chi_{k}\big\|_{L^{q}(\omega_{2})}\right\}^{p}\\ &+ C\sum\limits_{k=-\infty}^{\infty}[\omega_{1}(B_{k})]^{\frac{\alpha p}{n}}\left\{\sum\limits_{j=k-1}^{k+1}\big\|\mu_{\Omega}^{b}(f_{j})\chi_{k}\big\|_{L^{q}(\omega_{2})}\right\}^{p}\\ &+ C\sum\limits_{k=-\infty}^{\infty}[\omega_{1}(B_{k})]^{\frac{\alpha p}{n}}\left\{\sum\limits_{j=k+2}^{\infty}\big\|\mu_{\Omega}^{b}(f_{j})\chi_{k}\big\|_{L^{q}(\omega_{2})}\right\}^{p}\\ \triangleq& U_{1}+U_{2}+U_{3}. \end{aligned} $

首先考虑$U_{2}, $由引理2.1及引理2.4知

$ \begin{aligned} U_{2}\leq& C\big\|b\big\|_{{\hbox{BMO}}}^{p}\sum\limits_{k=-\infty}^{\infty}[\omega_{1}(B_{k})]^{\frac{\alpha p}{n}}\left\{\sum\limits_{j=k-1}^{k+1}\big\|f_{j}\big\|_{L^{q}(\omega_{2})}\right\}^{p}\\ \leq& C\big\|b\big\|_{{\hbox{BMO}}}^{p}\sum\limits_{k=-\infty}^{\infty}[\omega_{1}(B_{k})]^{\frac{\alpha p}{n}}\big\|f\chi_{k}\big\|_{L^{q}(\omega_{2})}^{p}\\ \leq& C\big\|b\big\|_{{\hbox{BMO}}}^{p}\big\|f\big\|_{K^{\alpha, p}_{q}(\omega_{1}, \omega_{2})}^{p}. \end{aligned} $

为了考虑$U_{1}, $我们首先估计$\big\|\mu_{\Omega}^{b}(f_{j})\chi_{k}\big\|_{L^{q}(\omega_{2})}, $有

$ \begin{aligned} &\big\|\mu_{\Omega}^{b}(f_{j})\chi_{k}\big\|_{L^{q}(\omega_{2})}= \left\{\int_{C_{k}}\left[\int_{0}^{\infty}\big|\int_{|x-y|\leq~t}\dfrac{\Omega(x, x-y)}{|x-y|^{n-1}}(b(x)-b(y))f_{j}(y)dy\big|^{2}\dfrac{dt}{t^{3}}\right]^{\frac{q}{2}}\\\omega_{2}(x)dx\right\}^{\frac{1}{q}}\\ =& \left\{\int_{C_{k}}\left[(\int_{0}^{|x|}+\int_{|x|}^{\infty})\big|\int_{|x-y|\leq~t}\dfrac{\Omega(x, x-y)}{|x-y|^{n-1}}(b(x)-b(y))f_{j}(y)dy\big|^{2}\dfrac{dt}{t^{3}}\right]^{\frac{q}{2}}\omega_{2}(x)dx\right\}^{\frac{1}{q}}\\ \leq& C\left\{\int_{C_{k}}\left[\int_{0}^{|x|}\big|\int_{|x-y|\leq~t}\dfrac{\Omega(x, x-y)}{|x-y|^{n-1}}(b(x)-b(y))f_{j}(y)dy\big|^{2}\dfrac{dt}{t^{3}}\right]^{\frac{q}{2}}\omega_{2}(x)dx\right\}^{\frac{1}{q}}\\ &+ C\left\{\int_{C_{k}}\left[\int_{|x|}^{\infty}\big|\int_{|x-y|\leq~t}\dfrac{\Omega(x, x-y)}{|x-y|^{n-1}}(b(x)-b(y))f_{j}(y)dy\big|^{2}\dfrac{dt}{t^{3}}\right]^{\frac{q}{2}}\omega_{2}(x)dx\right\}^{\frac{1}{q}}\\ \triangleq& C(A+B). \end{aligned} $

记$b_{k}=\dfrac{1}{|B_{k}|}\int_{B_{k}}|b(y)|dy, $首先考虑$\{\int_{C_{k}}|b(x)-b_{k}|^{q}\omega_{2}(x)dx\}^{\frac{1}{q}}.$取适当的$s, $使得$s>q, $且$\omega_{2}(x)$关于指数$\dfrac{s}{s-q}$满足反向Hölder不等式.由Hölder不等式及反向Hölder不等式得

$ \begin{aligned} &\left\{\int_{C_{k}}|b(x)-b_{k}|^{q}\omega_{2}(x)dx\right\}^{\frac{1}{q}}\leq C\left\{(\int_{C_{k}}|b(x)-b_{k}|^{s}dx)^{\frac{q}{s}}(\int_{C_{k}}(\omega_{2}(x))^{\frac{s}{s-q}}dx)^{\frac{s-q}{s}}\right\}^{\frac{1}{q}}\\ \leq& C|B_{k}|^{\frac{1}{s}}\left\{\dfrac{1}{|B_{k}|}\int_{B{k}}|b(x)-b_{k}|^{s}dx\right\}^{\frac{1}{s}}|B_{k}|^{\frac{1}{q}-\frac{1}{s}}\left\{\dfrac{1}{|B_{k}|}\int_{B_{k}}(\omega_{2}(x))^{\frac{s}{s-q}}dx\right\}^{\frac{s-q}{sq}}\\ \leq& C|B_{k}|^{\frac{1}{q}}\big\|b\big\|_{{\hbox{BMO}}}\left\{\dfrac{1}{|B_{k}|}\int_{B_{k}}\omega_{2}(x)dx\right\}^{\frac{1}{q}}\\ \leq& C|B_{k}|^{\frac{1}{q}}\big\|b\big\|_{{\hbox{BMO}}}\left[\dfrac{\omega_{2}(B_{k})}{|B_{k}|}\right]^{\frac{1}{q}}. \end{aligned} $

注意到$x\in C_{k}, y\in C_{j}, j\leq k-2, $则$|x-y|\sim|x|, $且$\big|\dfrac{1}{|x-y|^{2}}-\dfrac{1}{|x|^{2}}\big|\leq C\dfrac{|y|}{|x-y|^{3}}.$由$r'<q, $我们可选取适当的$\beta:-\frac{1}{r}<\beta<0, $使得$\alpha<-\beta-\frac{n}{q}+\frac{n}{r'}<n(\frac{1}{r'}-\frac{1}{q})+\frac{1}{r}, $由引理2.2及Minkowski不等式得

$ A=\left\{\int_{C_{k}}\left[\int_{0}^{|x|}\big|\int_{|x-y|\leq~t}\dfrac{\Omega(x, x-y)}{|x-y|^{n-1}}(b(x)-b(y))f_{j}(y)dy\big|^{2}\dfrac{dt}{t^{3}}\right]^{\frac{q}{2}}\omega_{2}(x)dx\right\}^{\frac{1}{q}}\\ \;\;\;\;\leq C\left\{\int_{C_{k}}\left[\int_{C_{j}}\dfrac{|\Omega(x, x-y)|}{|x-y|^{n-1}}|b(x)-b(y)||f_{j}(y)|\left(\int_{|x-y|\leq~t\leq|x|}\dfrac{dt}{t^{3}}\right)^{\frac{1}{2}}dy\right]^{q}\omega_{2}(x)dx\right\}^{\frac{1}{q}}\\ \;\;\;\;\leq C\left\{\int_{C_{k}}\left[\int_{C_{j}}\dfrac{|\Omega(x, x-y)|}{|x-y|^{n-1}}|b(x)-b(y)||f_{j}(y)|\big|\dfrac{1}{|x-y|^{2}}-\dfrac{1}{|x|^{2}}\big|^{\frac{1}{2}}dy\right]^{q}\omega_{2}(x)dx\right\}^{\frac{1}{q}}\\ \;\;\;\;\leq C\left\{\int_{C_{k}}\left[\int_{C_{j}}|\Omega(x, x-y)||b(x)-b(y)||f_{j}(y)|\dfrac{|y|^{\frac{1}{2}}}{|x|^{n+\frac{1}{2}}}dy\right]^{q}\omega_{2}(x)dx\right\}^{\frac{1}{q}}\\ \;\;\;\;\leq C\left\{\int_{C_{k}}\left[\int_{C_{j}}|\Omega(x, x-y)||y|^{\beta}|b(x)-b(y)||f_{j}(y)|\dfrac{|y|^{\frac{1}{2}-\beta}}{|x|^{n+\frac{1}{2}}}dy\right]^{q}\omega_{2}(x)dx\right\}^{\frac{1}{q}}\\ \;\;\;\;\leq C2^{\frac{j}{2}-\frac{k}{2}-kn-j\beta}\left\{\int_{C_{k}}\left[\int_{C_{j}}|\Omega(x, x-y)||y|^{\beta}|b(x)-b(y)||f_{j}(y)|dy\right]^{q}\omega_{2}(x)dx\right\}^{\frac{1}{q}}\\ \;\;\;\;\leq C2^{\frac{j}{2}-\frac{k}{2}-kn-j\beta}\left\{\int_{C_{k}}\left[\left(\int_{C_{j}}|\Omega(x, x-y)|^{r}|y|^{r\beta}dy\right)^{\frac{q}{r}}\\\;\;\;\;\left(\int_{C_{j}}|b(x)-b(y)|^{r'}|f_{j}(y)|^{r'}dy\right)^{\frac{q}{r'}}\right]\omega_{2}(x)dx\right\}^{\frac{1}{q}}\\ \;\;\;\;\leq C2^{\frac{j}{2}-\frac{k}{2}-kn-j\beta}\cdot2^{k(\beta+\frac{n}{r})}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\left\{\int_{C_{k}}\left(\int_{C_{j}}|b(x)-b(y)|^{r'}|f_{j}(y)|^{r'}dy\right)^{\frac{q}{r'}}\\\;\;\;\;\omega_{2}(x)dx\right\}^{\frac{1}{q}}\\ \;\;\;\;\leq C2^{\frac{j}{2}-\frac{k}{2}-kn-j\beta}\cdot2^{k(\beta+\frac{n}{r})}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})} \\ \;\;\;\;\left\{\int_{C_{k}} \left[\left(\int_{C_{j}}|b(x)-b(y)|^{\frac{qr'}{q-r'}}dy\right)^{\frac{q-r'}{r'}}\left(\int_{C_{j}}|f_{j}(y)|^{q}dy\right)\right]\omega_{2}(x)dx\right\}^{\frac{1}{q}} $

$ \leq C2^{\frac{j}{2}-\frac{k}{2}-kn-j\beta}\cdot2^{k(\beta+\frac{n}{r})}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\big\|f_{j}\big\|_{L^{q}({\mathbb{R}}^{n})}\left\{\int_{C_{k}}\left[\int_{C_{j}}|b(x)-b(y)|^{\frac{qr'}{q-r'}}dy\right]^{\frac{q-r'}{r'}}\\\;\;\;\;\omega_{2}(x)dx\right\}^{\frac{1}{q}}\\ \leq C2^{\frac{j}{2}-\frac{k}{2}-kn-j\beta}\cdot2^{k(\beta+\frac{n}{r})}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\big\|f_{j}\big\|_{L^{q}(\mathbb{R}^{n})}\left(\int_{C_{k}}\left[\int_{C_{j}}|b(x)-b_{k}|^{\frac{qr'}{q-r'}}dy\right]^{\frac{q-r'}{r'}}\\\;\;\;\;\omega_{2}(x)dx\right)^{\frac{1}{q}}\\ \;\;\;\;+ C2^{\frac{j}{2}-\frac{k}{2}-kn-j\beta}\cdot2^{k(\beta+\frac{n}{r})}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\big\|f_{j}\big\|_{L^{q}(\mathbb{R}^{n})}\left(\int_{C_{k}}\left[\int_{C_{j}}|b_{k}-b_{j}|^{\frac{qr'}{q-r'}}dy\right]^{\frac{q-r'}{r'}}\\\;\;\;\;\omega_{2}(x)dx\right)^{\frac{1}{q}}\\ \;\;\;\;+ C2^{\frac{j}{2}-\frac{k}{2}-kn-j\beta}\cdot2^{k(\beta+\frac{n}{r})}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\big\|f_{j}\big\|_{L^{q}(\mathbb{R}^{n})}\left(\int_{C_{k}}\left[\int_{C_{j}}|b(y)-b_{j}|^{\frac{qr'}{q-r'}}dy\right]^{\frac{q-r'}{r'}}\\\;\;\;\;\omega_{2}(x)dx\right)^{\frac{1}{q}}\\ \triangleq C(A_{1}+A_{2}+A_{3}). $

对$A_{1}, $有

$ A_{1}= 2^{\frac{j}{2}-\frac{k}{2}-kn-j\beta}\cdot2^{k(\beta+\frac{n}{r})}\\ \;\;\;\;\;\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\big\|f_{j}\big\|_{L^{q}({\mathbb{R}}^{n})}\left\{\int_{C_{k}}\left[\int_{C_{j}}|b(x)-b_{k}|^{\frac{qr'}{q-r'}}dy\right]^{\frac{q-r'}{r'}}\omega_{2}(x)dx\right\}^{\frac{1}{q}}\\ \\\;\;\;\;\leq C2^{\frac{j}{2}-\frac{k}{2}-kn-j\beta}\cdot2^{k(\beta+\frac{n}{r})}\cdot2^{jn(\frac{1}{r'}-\frac{1}{q})}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\big\|f_{j}\big\|_{L^{q}({\mathbb{R}}^{n})}\\\;\;\;\;\left\{\int_{C_{k}}|b(x)-b_{k}|^{q}\omega_{2}(x)dx\right\}^{\frac{1}{q}}\\ \\\;\;\;\;\leq C2^{\frac{j}{2}-\frac{k}{2}-kn-j\beta}\cdot2^{k(\beta+\frac{n}{r})}\cdot2^{jn(\frac{1}{r'}-\frac{1}{q})}\cdot2^{\frac{kn}{q}}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\big\|f_{j}\big\|_{L^{q}({\mathbb{R}}^{n})}\\\;\;\;\;\big\|b\big\|_{{\hbox{BMO}}}\left[\dfrac{\omega_{2}(B_{k})}{|B_{k}|}\right]^{\frac{1}{q}}\\ \\\;\;\;\;\leq C2^{\frac{j}{2}-\frac{k}{2}-kn-j\beta}\cdot2^{k(\beta+\frac{n}{r})}\\ \;\;\;\;\;\;\cdot2^{jn(\frac{1}{r'}-\frac{1}{q})}\cdot2^{\frac{kn}{q}}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\big\|f_{j}\big\|_{L^{q}(\omega_{2})}\big\|b\big\|_{{\hbox{BMO}}}\left[\dfrac{|B_{j}|}{\omega_{2}(B_{j})}\right]^{\frac{1}{q}}\left[\dfrac{\omega_{2}(B_{k})}{|B_{k}|}\right]^{\frac{1}{q}}\\ \;\;\;\;\leq C2^{(k-j)(\beta-\frac{1}{2}+\frac{n}{q}-\frac{n}{r'})}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\big\|b\big\|_{{\hbox{BMO}}}\big\|f_{j}\big\|_{L^{q}(\omega_{2})}. $

对$A_{2}, $注意到$|b_{k}-b_{j}|\leq C(k-j)\big\|b\big\|_{{\hbox{BMO}}}, $有

$ A_{2}= 2^{\frac{j}{2}-\frac{k}{2}-kn-j\beta}\cdot2^{k(\beta+\frac{n}{r})}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\big\|f_{j}\big\|_{L^{q}({\mathbb{R}}^{n})}\left\{\int_{C_{k}}\left[\int_{C_{j}}|b_{k}-b_{j}|^{\frac{qr'}{q-r'}}dy\right]^{\frac{q-r'}{r'}}\\\;\;\;\;\omega_{2}(x)dx\right\}^{\frac{1}{q}}\\ \;\;\;\;\leq C(k-j)2^{\frac{j}{2}-\frac{k}{2}-kn-j\beta}\cdot2^{k(\beta+\frac{n}{r})}\\ \;\;\;\;\cdot2^{jn(\frac{1}{r'}-\frac{1}{q})}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\big\|f_{j}\big\|_{L^{q}({\mathbb{R}}^{n})}\big\|b\big\|_{{\hbox{BMO}}}\left\{\int_{C_{k}}\omega_{2}(x)dx\right\}^{\frac{1}{q}}\\ \;\;\;\;\leq C(k-j)2^{\frac{j}{2}-\frac{k}{2}-kn-j\beta}\cdot2^{k(\beta+\frac{n}{r})}\cdot2^{jn(\frac{1}{r'}-\frac{1}{q})}\\ \;\;\;\;\cdot2^{\frac{kn}{q}}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\big\|f_{j}\big\|_{L^{q}(\omega_{2})}\big\|b\big\|_{{\hbox{BMO}}}\left[\dfrac{|B_{j}|}{\omega_{2}(B_{j})}\right]^{\frac{1}{q}}\left[\dfrac{\omega_{2}(B_{k})}{|B_{k}|}\right]^{\frac{1}{q}}\\ \;\;\;\;\leq C(k-j)2^{(k-j)(\beta-\frac{1}{2}+\frac{n}{q}-\frac{n}{r'})}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\big\|b\big\|_{{\hbox{BMO}}}\big\|f_{j}\big\|_{L^{q}(\omega_{2})}. $

对$A_{3}, $有

$ A_{3}= 2^{\frac{j}{2}-\frac{k}{2}-kn-j\beta}\cdot2^{k(\beta+\frac{n}{r})}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\big\|f_{j}\big\|_{L^{q}({\mathbb{R}}^{n})}\left\{\int_{C_{k}}\left[\int_{C_{j}}|b(y)-b_{j}|^{\frac{qr'}{q-r'}}dy\right]^{\frac{q-r'}{r'}}\\\;\;\;\;\omega_{2}(x)dx\right\}^{\frac{1}{q}}\\ \;\;\;\;\leq C2^{\frac{j}{2}-\frac{k}{2}-kn-j\beta}\cdot2^{k(\beta+\frac{n}{r})}\cdot2^{jn(\frac{1}{r'}-\frac{1}{q})}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\big\|f_{j}\big\|_{L^{q}({\mathbb{R}}^{n})}\big\|b\big\|_{{\hbox{BMO}}}\left\{\int_{C_{k}}\\\;\;\;\;\omega_{2}(x)dx\right\}^{\frac{1}{q}}\\ \;\;\;\;\leq C2^{\frac{j}{2}-\frac{k}{2}-kn-j\beta}\cdot2^{k(\beta+\frac{n}{r})}\\ \;\;\;\;\cdot2^{jn(\frac{1}{r'}-\frac{1}{q})}\cdot2^{\frac{kn}{q}}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\big\|f_{j}\big\|_{L^{q}(\omega_{2})}\big\|b\big\|_{{\hbox{BMO}}}\left[\dfrac{|B_{j}|}{\omega_{2}(B_{j})}\right]^{\frac{1}{q}}\left[\dfrac{\omega_{2}(B_{k})}{|B_{k}|}\right]^{\frac{1}{q}}\\ \;\;\;\;\leq C2^{(k-j)(\beta-\frac{1}{2}+\frac{n}{q}-\frac{n}{r'})}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\big\|b\big\|_{{\hbox{BMO}}}\big\|f_{j}\big\|_{L^{q}(\omega_{2})}. $

从而

$ A\leq C(k-j)2^{(k-j)(\beta-\frac{1}{2}+\frac{n}{q}-\frac{n}{r'})}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\big\|b\big\|_{BMO}\big\|f_{j}\big\|_{L^{q}(\omega_{2})}. $

对$B, $类似于$A$的估计方法, 得

$ B= \left\{\int_{C_{k}}\left[\int_{|x|}^{\infty}\big|\int_{|x-y|\leq~t}\dfrac{\Omega(x, x-y)}{|x-y|^{n-1}}(b(x)-b(y))f_{j}(y)dy\big|^{2}\dfrac{dt}{t^{3}}\right]^{\frac{q}{2}}\omega_{2}(x)dx\right\}^{\frac{1}{q}}\\ \;\;\;\;\leq C\left\{\int_{C_{k}}\left[\int_{C_{j}}\dfrac{|\Omega(x, x-y)|}{|x-y|^{n-1}}|b(x)-b(y)||f_{j}(y)|\left(\int_{|x-y|\leq~t, |x|\leq~t}\dfrac{dt}{t^{3}}\right)^{\frac{1}{2}}dy\right]^{q}\omega_{2}(x)dx\right\}^{\frac{1}{q}}\\ \;\;\;\;\leq C\left\{\int_{C_{k}}\left[\int_{C_{j}}\dfrac{|\Omega(x, x-y)|}{|x-y|^{n-1}}|b(x)-b(y)||f_{j}(y)|\dfrac{1}{|x-y|}dy\right]^{q}\omega_{2}(x)dx\right\}^{\frac{1}{q}}\\ \;\;\;\;\leq C\left\{\int_{C_{k}}\left[\int_{C_{j}}|\Omega(x, x-y)||b(x)-b(y)||f_{j}(y)|\dfrac{1}{|x|^{n}}dy\right]^{q}\omega_{2}(x)dx\right\}^{\frac{1}{q}}\\ \;\;\;\;\leq C\left\{\int_{C_{k}}\left[\int_{C_{j}}|\Omega(x, x-y)||y|^{\beta}|b(x)-b(y)||f_{j}(y)|\dfrac{|y|^{-\beta}}{|x|^{n}}dy\right]^{q}\omega_{2}(x)dx\right\}^{\frac{1}{q}}\\ \;\;\;\;\leq C2^{-kn-j\beta}\left\{\int_{C_{k}}\left[\left(\int_{C_{j}}|\Omega(x, x-y)|^{r}|y|^{r\beta}dy\right)^{\frac{q}{r}}\left(\int_{C_{j}}|b(x)-b(y)|^{r'}|f_{j}(y)|^{r'}dy\right)^{\frac{q}{r'}}\right]\\\;\;\;\;\omega_{2}(x)dx\right\}^{\frac{1}{q}}\\ \;\;\;\;\leq C2^{-kn-j\beta}\cdot2^{k(\beta+\frac{n}{r})}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\left\{\int_{C_{k}}\left(\int_{C_{j}}|b(x)-b(y)|^{r'}|f_{j}(y)|^{r'}dy\right)^{\frac{q}{r'}}\\\;\;\;\;\omega_{2}(x)dx\right\}^{\frac{1}{q}}\\ \;\;\;\;\leq C2^{-kn-j\beta}\cdot2^{k(\beta+\frac{n}{r})}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\\ \;\;\;\;\left\{\int_{C_{k}}\left[\left(\int_{C_{j}}|b(x)-b(y)|^{\frac{qr'}{q-r'}}dy\right)^{\frac{q-r'}{r'}}\left(\int_{C_{j}}|f_{j}(y)|^{q}dy\right)\right]\omega_{2}(x)dx\right\}^{\frac{1}{q}}\\ \;\;\;\;\leq C2^{-kn-j\beta}\cdot2^{k(\beta+\frac{n}{r})}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\big\|f_{j}\big\|_{L^{q}({\mathbb{R}}^{n})}\left\{\int_{C_{k}}\left[\int_{C_{j}}|b(x)-b(y)|^{\frac{qr'}{q-r'}}dy\right]^{\frac{q-r'}{r'}}\\\;\;\;\;\omega_{2}(x)dx\right\}^{\frac{1}{q}} $

$ \leq C2^{-kn-j\beta}\cdot2^{k(\beta+\frac{n}{r})}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\big\|f_{j}\big\|_{L^{q}({\mathbb{R}}^{n})}\left(\int_{C_{k}}\left[\int_{C_{j}}|b(x)-b_{k}|^{\frac{qr'}{q-r'}}dy\right]^{\frac{q-r'}{r'}}\\\;\;\;\;\omega_{2}(x)dx\right)^{\frac{1}{q}}\\ \;\;\;\;+ C2^{-kn-j\beta}\cdot2^{k(\beta+\frac{n}{r})}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\big\|f_{j}\big\|_{L^{q}({\mathbb{R}}^{n})}\left(\int_{C_{k}}\left[\int_{C_{j}}|b_{k}-b_{j}|^{\frac{qr'}{q-r'}}dy\right]^{\frac{q-r'}{r'}}\\\;\;\;\;\omega_{2}(x)dx\right)^{\frac{1}{q}}\\ \;\;\;\;+ C2^{-kn-j\beta}\cdot2^{k(\beta+\frac{n}{r})}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\big\|f_{j}\big\|_{L^{q}({\mathbb{R}}^{n})}\left(\int_{C_{k}}\left[\int_{C_{j}}|b(y)-b_{j}|^{\frac{qr'}{q-r'}}dy\right]^{\frac{q-r'}{r'}}\\\;\;\;\;\omega_{2}(x)dx\right)^{\frac{1}{q}}\\ \triangleq C(B_{1}+B_{2}+B_{3}). $

对$B_{1}, $有

$ B_{1}= 2^{-kn-j\beta}\cdot2^{k(\beta+\frac{n}{r})}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\big\|f_{j}\big\|_{L^{q}({\mathbb{R}}^{n})}\\ \;\;\;\;\left\{\int_{C_{k}}\left[\int_{C_{j}}|b(x)-b_{k}|^{\frac{qr'}{q-r'}}dy\right]^{\frac{q-r'}{r'}}\omega_{2}(x)dx\right\}^{\frac{1}{q}}\\ \;\;\;\;\leq C2^{-kn-j\beta}\cdot2^{k(\beta+\frac{n}{r})}\cdot2^{jn(\frac{1}{r'}-\frac{1}{q})}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\big\|f_{j}\big\|_{L^{q}({\mathbb{R}}^{n})}\left\{\int_{C_{k}}|b(x)-b_{k}|^{q}\\\;\;\;\;\omega_{2}(x)dx\right\}^{\frac{1}{q}}\\ \;\;\;\;\leq C2^{-kn-j\beta}\cdot2^{k(\beta+\frac{n}{r})}\cdot2^{jn(\frac{1}{r'}-\frac{1}{q})}\cdot2^{\frac{kn}{q}}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\big\|f_{j}\big\|_{L^{q}({\mathbb{R}}^{n})}\big\|b\big\|_{{\hbox{BMO}}}\left[\dfrac{\omega_{2}(B_{k})}{|B_{k}|}\right]^{\frac{1}{q}}\\ \;\;\;\;\leq C2^{-kn-j\beta}\cdot2^{k(\beta+\frac{n}{r})}\cdot2^{jn(\frac{1}{r'}-\frac{1}{q})}\\ \;\;\;\;\cdot2^{\frac{kn}{q}}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\big\|f_{j}\big\|_{L^{q}(\omega_{2})}\big\|b\big\|_{{\hbox{BMO}}}\left[\dfrac{|B_{j}|}{\omega_{2}(B_{j})}\right]^{\frac{1}{q}}\left[\dfrac{\omega_{2}(B_{k})}{|B_{k}|}\right]^{\frac{1}{q}}\\ \;\;\;\;\leq C2^{(k-j)(\beta+\frac{n}{q}-\frac{n}{r'})}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\big\|b\big\|_{{\hbox{BMO}}}\big\|f_{j}\big\|_{L^{q}(\omega_{2})}. $

对$B_{2}, $注意到$|b_{k}-b_{j}|\leq C(k-j)\big\|b\big\|_{{\hbox{BMO}}}, $有

$ B_{2}= 2^{-kn-j\beta}\cdot2^{k(\beta+\frac{n}{r})}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\big\|f_{j}\big\|_{L^{q}({\mathbb{R}}^{n})}\left\{\int_{C_{k}}\left[\int_{C_{j}}|b_{k}-b_{j}|^{\frac{qr'}{q-r'}}dy\right]^{\frac{q-r'}{r'}}\\\;\;\;\;\omega_{2}(x)dx\right\}^{\frac{1}{q}}\\ \;\;\;\;\leq C(k-j)2^{-kn-j\beta}\cdot2^{k(\beta+\frac{n}{r})}\cdot2^{jn(\frac{1}{r'}-\frac{1}{q})}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\big\|f_{j}\big\|_{L^{q}({\mathbb{R}}^{n})}\big\|b\big\|_{{\hbox{BMO}}}\left\{\int_{C_{k}}\\\;\;\;\;\omega_{2}(x)dx\right\}^{\frac{1}{q}}\\ \;\;\;\;\leq C(k-j)2^{-kn-j\beta}\cdot2^{k(\beta+\frac{n}{r})}\cdot2^{jn(\frac{1}{r'}-\frac{1}{q})}\\ \;\;\;\;\cdot2^{\frac{kn}{q}}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\big\|f_{j}\big\|_{L^{q}(\omega_{2})}\big\|b\big\|_{{\hbox{BMO}}}\left[\dfrac{|B_{j}|}{\omega_{2}(B_{j})}\right]^{\frac{1}{q}}\left[\dfrac{\omega_{2}(B_{k})}{|B_{k}|}\right]^{\frac{1}{q}}\\ \;\;\;\;\leq C(k-j)2^{(k-j)(\beta+\frac{n}{q}-\frac{n}{r'})}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\big\|b\big\|_{{\hbox{BMO}}}\big\|f_{j}\big\|_{L^{q}(\omega_{2})}. $

对$B_{3}, $有

$ B_{3}= 2^{-kn-j\beta}\cdot2^{k(\beta+\frac{n}{r})}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\big\|f_{j}\big\|_{L^{q}({\mathbb{R}}^{n})}\left\{\int_{C_{k}}\left[\int_{C_{j}}|b(y)-b_{j}|^{\frac{qr'}{q-r'}}dy\right]^{\frac{q-r'}{r'}}\\\;\;\;\;\omega_{2}(x)dx\right\}^{\frac{1}{q}}\\ \;\;\;\;\leq C2^{-kn-j\beta}\cdot2^{k(\beta+\frac{n}{r})}\cdot2^{jn(\frac{1}{r'}-\frac{1}{q})}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\big\|f_{j}\big\|_{L^{q}({\mathbb{R}}^{n})}\big\|b\big\|_{{\hbox{BMO}}}\left\{\int_{C_{k}}\omega_{2}(x)dx\right\}^{\frac{1}{q}}\\ $

$ \begin{aligned} \leq& C2^{-kn-j\beta}\cdot2^{k(\beta+\frac{n}{r})}\cdot2^{jn(\frac{1}{r'}-\frac{1}{q})}\\ &\cdot2^{\frac{kn}{q}}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\big\|f_{j}\big\|_{L^{q}(\omega_{2})}\big\|b\big\|_{{\hbox{BMO}}}\left[\dfrac{|B_{j}|}{\omega_{2}(B_{j})}\right]^{\frac{1}{q}}\left[\dfrac{\omega_{2}(B_{k})}{|B_{k}|}\right]^{\frac{1}{q}}\\ \leq& C2^{(k-j)(\beta+\frac{n}{q}-\frac{n}{r'})}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\big\|b\big\|_{{\hbox{BMO}}}\big\|f_{j}\big\|_{L^{q}(\omega_{2})}. \end{aligned} $

从而

$ B\leq C(k-j)2^{(k-j)(\beta+\frac{n}{q}-\frac{n}{r'})}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\big\|b\big\|_{{\hbox{BMO}}}\big\|f_{j}\big\|_{L^{q}(\omega_{2})}. $

故根据引理2.1(i), 可得

$ U_{1}\leq C\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}^{p}\big\|b\big\|_{{\hbox{BMO}}}^{p}\sum\limits_{k=-\infty}^{\infty}[\omega_{1}(B_{k})]^{\frac{\alpha p}{n}}\left\{\sum\limits_{j=-\infty}^{k-2}(k-j)2^{(k-j)(\beta-\frac{1}{2}+\frac{n}{q}-\frac{n}{r'})}\\\;\;\;\;\big\|f_{j}\big\|_{L^{q}(\omega_{2})}\right\}^{p}\\ \;\;\;\;+ C\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}^{p}\big\|b\big\|_{{\hbox{BMO}}}^{p}\sum\limits_{k=-\infty}^{\infty}[\omega_{1}(B_{k})]^{\frac{\alpha p}{n}}\left\{\sum\limits_{j=-\infty}^{k-2}(k-j)2^{(k-j)(\beta+\frac{n}{q}-\frac{n}{r'})}\;\;\;\;\big\|f_{j}\big\|_{L^{q}(\omega_{2})}\right\}^{p}\\ \;\;\;\;\leq C\big\|b\big\|_{{\hbox{BMO}}}^{p}\sum\limits_{k=-\infty}^{\infty}\left\{\sum\limits_{j=-\infty}^{k-2}(k-j)2^{(k-j)(\beta+\frac{n}{q}-\frac{n}{r'})}\left[\dfrac{\omega_{1}(B_{k})}{\omega_{1}(B_{j})}\right]^{\frac{\alpha}{n}}[\omega_{1}(B_{j})]^{\frac{\alpha}{n}}\big\|f_{j}\big\|_{L^{q}(\omega_{2})}\right\}^{p}\\ \;\;\;\;\leq C\big\|b\big\|_{{\hbox{BMO}}}^{p}\sum\limits_{k=-\infty}^{\infty}\left\{\sum\limits_{j=-\infty}^{k-2}(k-j)2^{(k-j)(\alpha+\beta+\frac{n}{q}-\frac{n}{r'})}[\omega_{1}(B_{j})]^{\frac{\alpha}{n}}\big\|f_{j}\big\|_{L^{q}(\omega_{2})}\right\}^{p}. $

当$0<p\leq1$时, 利用$(\sum\limits_{j=-\infty}^{\infty}|a_{j}|)^{p}\leq\sum\limits_{j=-\infty}^{\infty}|a_{j}|^{p}, $可得

$ \begin{aligned} U_{1}\leq& C\big\|b\big\|_{{\hbox{BMO}}}^{p}\sum\limits_{k=-\infty}^{\infty}\left\{\sum\limits_{j=-\infty}^{k-2}(k-j)^{p}2^{p(k-j)(\alpha+\beta+\frac{n}{q}-\frac{n}{r'})}[\omega_{1}(B_{j})]^{\frac{\alpha p}{n}}\big\|f_{j}\big\|_{L^{q}(\omega_{2})}^{p}\right\}\\ \leq& C\big\|b\big\|_{{\hbox{BMO}}}^{p}\sum\limits_{j=-\infty}^{\infty}[\omega_{1}(B_{j})]^{\frac{\alpha p}{n}}\big\|f_{j}\big\|_{L^{q}(\omega_{2})}^{p}\left\{\sum\limits_{k=j+2}^{\infty}(k-j)^{p}2^{p(k-j)(\alpha+\beta+\frac{n}{q}-\frac{n}{r'})}\right\}\\ \leq& C\big\|b\big\|_{{\hbox{BMO}}}^{p}\big\|f\big\|_{K^{\alpha, p}_{q}(\omega_{1}, \omega_{2})}^{p}. \end{aligned} $

当$1<p<\infty$时, 利用Hölder不等式, 可得

$ \begin{aligned} U_{1}\leq& C\big\|b\big\|_{{\hbox{BMO}}}^{p}\sum\limits_{k=-\infty}^{\infty}\left\{\sum\limits_{j=-\infty}^{k-2}(k-j)^{p}2^{p(k-j)(\alpha+\beta+\frac{n}{q}-\frac{n}{r'})/2}[\omega_{1}(B_{j})]^{\frac{\alpha p}{n}}\big\|f_{j}\big\|_{L^{q}(\omega_{2})}^{p}\right\}\\ &\times \left\{\sum\limits_{j=-\infty}^{k-2}2^{p'(k-j)(\alpha+\beta+\frac{n}{q}-\frac{n}{r'})/2}\right\}^{\frac{p}{p'}} \leq C\big\|b\big\|_{{\hbox{BMO}}}^{p}\big\|f\big\|_{K^{\alpha, p}_{q}(\omega_{1}, \omega_{2})}^{p}. \end{aligned} $

最后考虑$U_{3}, $首先估计$\big\|\mu_{\Omega}^{b}(f_{j})\chi_{k}\big\|_{L^{q}(\omega_{2})}, $有

$ \big\|\mu_{\Omega}^{b}(f_{j})\chi_{k}\big\|_{L^{q}(\omega_{2})}= \;\;\;\;\left\{\int_{C_{k}}\left[\int_{0}^{\infty}\big|\int_{|x-y|\leq~t}\dfrac{\Omega(x, x-y)}{|x-y|^{n-1}}(b(x)-b(y))f_{j}(y)dy\big|^{2}\dfrac{dt}{t^{3}}\right]^{\frac{q}{2}}\\\;\;\;\;\omega_{2}(x)dx\right\}^{\frac{1}{q}}\\ \;\;\;\;= \left\{\int_{C_{k}}\left[(\int_{0}^{|y|}+\int_{|y|}^{\infty})\big|\int_{|x-y|\leq~t}\dfrac{\Omega(x, x-y)}{|x-y|^{n-1}}(b(x)-b(y))f_{j}(y)dy\big|^{2}\dfrac{dt}{t^{3}}\right]^{\frac{q}{2}}\omega_{2}(x)dx\right\}^{\frac{1}{q}}\\ \;\;\;\;\leq C\left\{\int_{C_{k}}\left[\int_{0}^{|y|}\big|\int_{|x-y|\leq~t}\dfrac{\Omega(x, x-y)}{|x-y|^{n-1}}(b(x)-b(y))f_{j}(y)dy\big|^{2}\dfrac{dt}{t^{3}}\right]^{\frac{q}{2}}\omega_{2}(x)dx\right\}^{\frac{1}{q}}\\ $

$ \begin{aligned} &+ C\left\{\int_{C_{k}}\left[\int_{|y|}^{\infty}\big|\int_{|x-y|\leq~t}\dfrac{\Omega(x, x-y)}{|x-y|^{n-1}}(b(x)-b(y))f_{j}(y)dy\big|^{2}\dfrac{dt}{t^{3}}\right]^{\frac{q}{2}}\omega_{2}(x)dx\right\}^{\frac{1}{q}}\\ \triangleq& C(A'+B'). \end{aligned} $

注意到$x\in C_{k}, y\in C_{j}, j\geq k+2, $则$|x-y|\sim|y|, $且$\big|\dfrac{1}{|x-y|^{2}}-\dfrac{1}{|y|^{2}}\big|\leq C\dfrac{|x|}{|x-y|^{3}}.$由$r'<q, $我们可选取适当的$\beta:-n-\frac{1}{r}<\beta<-n, $使得$\alpha>-\frac{\beta}{\delta}-\frac{n}{q}-\frac{n}{\delta r}>n(\frac{1}{\delta r'}-\frac{1}{q}), $类似于$A$与$B$的估计方法, 由引理2.3及Minkowski不等式得

$ \begin{eqnarray*}&& A'\leq C(j-k)2^{(k-j)(\beta+\frac{1}{2}+\frac{n}{r}+\delta\frac{n}{q})}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\big\|b\big\|_{BMO}\big\|f_{j}\big\|_{L^{q}(\omega_{2})}, \\ && B'\leq C(j-k)2^{(k-j)(\beta+\frac{n}{r}+\delta\frac{n}{q})}\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}\big\|b\big\|_{BMO}\big\|f_{j}\big\|_{L^{q}(\omega_{2})}.\end{eqnarray*} $

故由引理2.1(ⅱ), 可得

$ U_{3}\leq C\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}^{p}\big\|b\big\|_{BMO}^{p}\sum\limits_{k=-\infty}^{\infty}[\omega_{1}(B_{k})]^{\frac{\alpha p}{n}}\left\{\sum\limits_{j=k+2}^{\infty}(j-k)2^{(k-j)(\beta+\frac{1}{2}+\frac{n}{r}+\delta\frac{n}{q})}\\\;\;\;\;\big\|f_{j}\big\|_{L^{q}(\omega_{2})}\right\}^{p}\\ \;\;\;\;+ C\big\|\Omega\big\|_{L^{\infty}({\mathbb{R}}^{n})\times L^{r}(S^{n-1})}^{p}\big\|b\big\|_{BMO}^{p}\sum\limits_{k=-\infty}^{\infty}[\omega_{1}(B_{k})]^{\frac{\alpha p}{n}}\left\{\sum\limits_{j=k+2}^{\infty}(j-k)2^{(k-j)(\beta+\frac{n}{r}+\delta\frac{n}{q})}\\\;\;\;\;\big\|f_{j}\big\|_{L^{q}(\omega_{2})}\right\}^{p}\\ \;\;\;\;\leq C\big\|b\big\|_{{\hbox{BMO}}}^{p}\sum\limits_{k=-\infty}^{\infty}\left\{\sum\limits_{j=k+2}^{\infty}(j-k)2^{(k-j)(\beta+\frac{n}{r}+\delta\frac{n}{q})}\left[\dfrac{\omega_{1}(B_{k})}{\omega_{1}(B_{j})}\right]^{\frac{\alpha}{n}}[\omega_{1}(B_{j})]^{\frac{\alpha}{n}}\big\|f_{j}\big\|_{L^{q}(\omega_{2})}\right\}^{p}\\ \;\;\;\;\leq C\big\|b\big\|_{{\hbox{BMO}}}^{p}\sum\limits_{k=-\infty}^{\infty}\left\{\sum\limits_{j=k+2}^{\infty}(j-k)2^{(k-j)(\delta\alpha+\beta+\frac{n}{r}+\delta\frac{n}{q})}[\omega_{1}(B_{j})]^{\frac{\alpha}{n}}\big\|f_{j}\big\|_{L^{q}(\omega_{2})}\right\}^{p}. $

当$0<p\leq1$时, 利用$(\sum\limits_{j=-\infty}^{\infty}|a_{j}|)^{p}\leq\sum\limits_{j=-\infty}^{\infty}|a_{j}|^{p}, $可得

$ \begin{aligned} U_{3}\leq& C\big\|b\big\|_{{\hbox{BMO}}}^{p}\sum\limits_{k=-\infty}^{\infty}\left\{\sum\limits_{j=k+2}^{\infty}(j-k)^{p}2^{p(k-j)(\delta\alpha+\beta+\frac{n}{r}+\delta\frac{n}{q})}[\omega_{1}(B_{j})]^{\frac{\alpha p}{n}}\big\|f_{j}\big\|_{L^{q}(\omega_{2})}^{p}\right\}\\ \leq& C\big\|b\big\|_{{\hbox{BMO}}}^{p}\sum\limits_{j=-\infty}^{\infty}[\omega_{1}(B_{j})]^{\frac{\alpha p}{n}}\big\|f_{j}\big\|_{L^{q}(\omega_{2})}^{p}\left\{\sum\limits_{k=-\infty}^{j-2}(j-k)^{p}2^{p(k-j)(\delta\alpha+\beta+\frac{n}{r}+\delta\frac{n}{q})}\right\}\\ \leq& C\big\|b\big\|_{{\hbox{BMO}}}^{p}\big\|f\big\|_{K^{\alpha, p}_{q}(\omega_{1}, \omega_{2})}^{p}. \end{aligned} $

当$1<p<\infty$时, 利用Hölder不等式, 可得

$ \begin{aligned} U_{3}\leq& C\big\|b\big\|_{{\hbox{BMO}}}^{p}\sum\limits_{k=-\infty}^{\infty}\left\{\sum\limits_{j=k+2}^{\infty}(j-k)^{p}2^{p(k-j)(\delta\alpha+\beta+\frac{n}{r}+\delta\frac{n}{q})/2}[\omega_{1}(B_{j})]^{\frac{\alpha p}{n}}\big\|f_{j}\big\|_{L^{q}(\omega_{2})}^{p}\right\}\\ &\times \left\{\sum\limits_{j=k+2}^{\infty}2^{p'(k-j)(\delta\alpha+\beta+\frac{n}{r}+\delta\frac{n}{q})/2}\right\}^{\frac{p}{p'}} \leq C\big\|b\big\|_{{\hbox{BMO}}}^{p}\big\|f\big\|_{K^{\alpha, p}_{q}(\omega_{1}, \omega_{2})}^{p}. \end{aligned} $

从而完成了定理的证明.