在${\Bbb R}^d$中定义与位势$V$相关的Schrödinger算子为$L=-\Delta + V$,其中$\Delta\equiv\sum\limits_{j=1}^{d}\frac{\partial^{2}}{\partial x_{j}^{2}}.$ 与算子$L$相关的半群算子$T_{t}^{L}$定义为 \begin{equation}\label{eq:1} T_{t}^{L}f(x)=e^{-tL}f(x)=\int_{{\Bbb R}^d}K_{t}^{L}(x,y)f(y){\rm d}y, \tag{1.1}\end{equation} 其中$f\in {{L}^{2}}({{\mathbb{R}}^{d}}),t>0$.

设$1<q<\infty,V\in L_{loc}^{q}$,如果存在常数$C=C(q,V)>0$,使得对$B\in {\Bbb R}^d$,有 \begin{equation}\label{eq:1.1} \bigg(\frac{1}{|B|}\int_{B}V(x)^{q}{\rm d}x\bigg)^{\frac{1}{q}}\leq C\frac{1}{|B|}\int_{B}V(x){\rm d}x \tag{1.2}\end{equation} 成立,则称$V\in B_{q}({\Bbb R}^d)$. 易见对某些$\epsilon>0$,当$V\in B_{q}({\Bbb R}^d)$,则$V\in B_{q+\epsilon}({\Bbb R}^d)$.如果$q_{1}>q_{2}$,则$B_{q_{1}}({\Bbb R}^d)\subset B_{q_{2}}({\Bbb R}^d)$.

定义与$V$相关的辅助函数$\rho(x)=\rho(x,V)$如下 \begin{equation}\label{eq:1.2} \rho(x)=\sup_{r>0}\bigg\{r:\frac{1}{r^{d-2}}\int_{B(x,r)}V(y){\rm d}y\leq1\bigg\}, x\in {\Bbb R}^d. \tag{1.3}\end{equation}

而在文献^[1]和^[2]中提及的与Schrödinger算子相关的Campanato型空间 $\Lambda_{L}^{\beta,p}({\Bbb R}^d)$定义如下.

定义1.1  设$p\in (0,\infty)$,$\beta\in R$,如果存在非负常数$C$,使得对任意的$f\in \text{L}_{loc}^{p}({{\mathbb{R}}^{d}})$和$0<s<\rho(x)\leq r,$ 有 \[ \bigg\{\frac{1}{|B(x,s)|^{1+p\beta}}\int_{B(x,s)}|f(y)-f_{B(x,s)}|^{p}{\rm d}y\bigg\}^{\frac{1}{p}} +\bigg\{\frac{1}{|B(x,r)|^{1+p\beta}}\int_{B(x,r)}|f(y)|^{p}{\rm d}y\bigg\}^{\frac{1}{p}}\leq C \] 成立,则称$f\in\Lambda_{L}^{\beta,p}({\Bbb R}^d)$. 其中$f_{B}\equiv\frac{1}{|B|}\int_{B}f(y){\rm d}y$.我们把满足上式中最小的$C$称为$f$在$\Lambda_{L}^{\beta,p}({\Bbb R}^d)$上的范数,记作$||f||_{\Lambda_{L}^{\beta,p}({\Bbb R}^d)}.$

注意到当 $p\in [1,\infty)$时,${\Lambda_{L}^{0,p}({\Bbb R}^d)}\equiv BMO_{L}({\Bbb R}^d)$,相关性质可以参见文献^[3]. 由上述定义易证当 $\beta\in[0,\infty)$且$p_{1},p_{2}\in[1,\infty)$时,有$\Lambda_{L}^{ \beta,p_{1}}({\Bbb R}^d)\equiv\Lambda_{L}^{\beta,p_{2}}({\Bbb R}^d)$, 特别的$||f||_{\Lambda_{L}^{\beta,p_{1}}({\Bbb R}^d)}\sim||f||_{\Lambda_{L}^{\beta,p_{2}}({\Bbb R}^d)}.$ 而经典的Campanato空间定义为$\Lambda^{\frac{1}{p}-1}({\Bbb R}^d)$易证 $\Lambda_{L}^{\frac{1}{p}-1}({\Bbb R}^d)\subseteq\Lambda^{\frac{1}{p}-1}({\Bbb R}^d).$

对任意的$0<\alpha<d,x\in {\Bbb R}^d.$ 定义$I_{\alpha}^{L}$如下 \begin{equation}\label{eq:1.4} I_{\alpha}^{L}f(x)=L^{-\frac{\alpha}{2}}f(x)= \int_{0}^{\infty}T_{t}^{L}f(x)t^{-\frac{\alpha}{2}}\frac{{\rm d}t}{t} \tag{1.4}\end{equation} 在文献^[4]中江寅生证明了当$0<\alpha<d$,$d/(d+\delta)<p_{1}<1<p_{2}<\infty$且满足$1/p_{2}=1/p_{1}-\alpha/d$时,位势算子$I_{\alpha}^{L}$是$L^{p_{2}'}({\Bbb R}^d)$到$\Lambda_{L}^{\frac{1}{p_{1}}-1}({\Bbb R}^d)$有界的. 同时也证明了当$0<\alpha<1$,$d/(d+\delta)<p_{1}<p_{2}<1$且满足$1/p_{2}=1/p_{1}-\alpha/d$时,位势算子$I_{\alpha}^{L}$是$\Lambda_{L}^{1/{p_{2}}-1}({\Bbb R}^d)$到$\Lambda_{L}^{1/{p_{1}}-1}({\Bbb R}^d)$有界的. 受文献^[4]的启发,我们将研究$b\in BMO$时,与$L$算子相关的Riesz位势算子的交换子$[b,I_{\alpha}^{L}]$在Campanato型空间上的有界性. 全文总设$\frac{d}{2}<q<d(d\geq3)$,且$\delta=2-\frac{d}{q}$.

下面给出本文的主要研究结果.

定理1.1 设$0<\alpha<d$,$d/(d+\delta)<p_{1}<1<p_{2}<\infty$且满足$1/p_{2}=1/p_{1}-\alpha/d,$则存在常数$C>0$使得对任意的$b\in BMO$和$f\in L^{p_{2}'}({\Bbb R}^d)$,有 \[\|[b,I_{\alpha}^{L}]f\|_{\Lambda_{L}^{1/{p_{1}}-1}({\Bbb R}^d)}\leq C\|b\|_{BMO}\|f\|_{L^{p_{2}'}({\Bbb R}^d)},\] 其中$1/{p_{2}}+1/{p_{2}}'=1.$

定理1.2 设$0<\alpha<1$,$d/(d+\delta)<p_{1}<p_{2}<1$且满足$1/p_{2}=1/p_{1}-\alpha/d,$ 则存在常数$C>0$使得对任意的$b\in \text{BMO}$和$f\in \Lambda_{L}^{1/{p_{2}}-1}({\Bbb R}^d)$,交换子$[b,I_{\alpha}^{L}]$是 $\Lambda_{L}^{1/{p_{2}}-1}({\Bbb R}^d)$到$\Lambda_{L}^{1/{p_{1}}-1}({\Bbb R}^d)$有界的,即 \[\|[b,I_{\alpha}^{L}]f\|_{\Lambda_{L}^{1/{p_{1}}-1}({\Bbb R}^d)} \leq C\|b\|_{BMO}\|f\|_{\Lambda_{L}^{1/{p_{2}}-1}({\Bbb R}^d)}.\] 全文用$C$或$A$表示与主要参量无关的正常数,或者是用$C_{1}$和$A_{1}$表示某些不变的常数.

在这部分我们先来回顾下辅助函数$\rho$和半群算子的核$K_t(x)$的相关性质. 由文献^[6]知存在常数 $C_{0}>0$和$k_{0}\leq 1$,使得对任意的$x,y\in {\Bbb R}^d,$ 有 \begin{equation}\label{eq:1.5} \frac{1}{C_{0}}\bigg(1+\frac{|x-y|}{\rho(x)}\bigg)^{-k_{0}}\leq\frac{\rho(y)}{\rho(x)}\leq C_{0}\bigg(1+\frac{|x-y|}{\rho(x)}\bigg)^{\frac{k_{0}}{k_{0}+1}}. \tag{2.1}\end{equation}

特别的,当$y\in B(x,r)$和$r\leq C\rho(x)$时,$\rho(x)\sim\rho(y)$.

设$K_{t}^{L}$是${\Bbb R}^d$上热半群$\{T_{t}=e^{t\Delta}:t>0\}$上的卷积核,因为$V$是非负的,由Feynman-Kac公式可知 \begin{equation}\label{eq:1.6} 0\leq K_{t}^{L}(x,y)\leq K_{t}(x-y):=4\pi t^{-\frac{d}{2}}e^{-\frac{|x-y|^{2}}{4t}}. \tag{2.2}\end{equation}

由文献^[5]知,当$\delta>0$时,有 \begin{equation}\label{eq:1.7} 0\leq K_{t}(x-y)- K_{t}^{L}(x,y)\leq Ct^{-\frac{d}{2}}e^{-A\frac{|x-y|^{2}}{t}}\bigg(\frac{\sqrt{t}}{\rho(x)}\bigg)^{\delta}, \tag{2.3}\end{equation} 且对任意的$N>0$,存在常数$C_{N}>0$,使得 \begin{equation}\label{eq:1.8} 0\leq K_{t}^{L}(x,y)\leq C_{N}t^{-\frac{d}{2}}e^{-A\frac{|x-y|^{2}}{t}} \bigg(1+\frac{\sqrt{t}}{\rho(x)}+\frac{\sqrt{t}}{\rho(y)}\bigg)^{-N}, \tag{2.4}\end{equation} 而对任意的$N>0,$ 存在常数$C_{N}>0$使得对任意的$|h|\leq\sqrt{t}$,有 \begin{equation}\label{eq:1.9} |K_{t}^{L}(x+h,y)-K_{t}^{L}(x,y)|\leq C_{N}\bigg(\frac{|h|}{\sqrt{t}}\bigg)^\delta t^{-\frac{d}{2}} e^{-A\frac{|x-y|^{2}}{t}} \bigg(1+\frac{\sqrt{t}}{\rho(x)}+\frac{\sqrt{t}}{\rho(y)}\bigg)^{-N}.\tag{2.5}\end{equation}

为证明我们的结论,我们还需要下列引理.

引理2.1^[4] 设$f\in L_{loc}({\Bbb R}^d)$,则对任意的$x\in {\Bbb R}^d$和$t>0$,有 \begin{equation}\label{eq:1.10} \int_{{\Bbb R}^d}e^{-\frac{|x-y|^{2}}{t}}|f(y)|{\rm d}y\leq Ct^{\frac{d}{2}}Mf(x), \tag{2.6}\end{equation} 其中$M$是Hardy-Littlewood极大算子.

引理2.2^[4]  设$1<p\leq\infty,f\in L^{p}({\Bbb R}^d)$,则对任意的$x\in {\Bbb R}^d$和$t>0,$ 有 \begin{equation}\label{eq:1.13} \int_{\mathbb{R}^d}e^{-\frac{|x-y|^{2}}{t}}|f(y)|{\rm d}y\leq Ct^{\frac{d}{2p'}}\|f(x)\|_{L^{p}({\Bbb R}^d)}. \tag{2.7}\end{equation}

引理2.3 设$1<p<\infty$,则对任意的$f\in L^{p}({\Bbb R}^d)$和$b\in BMO({\Bbb R}^d)$,使得对任意的$x\in {\Bbb R}^d,t>0$, \begin{equation} \int_{{{\mathbb{R}}^{d}}}{{{e}^{-\frac{|x-y{{|}^{2}}}{t}}}}|(b(y)-{{b}_{B}})f(y)|\text{d}y\le C\|b{{\|}_{BMO}}{{t}^{\frac{d}{2{p}'}}}(1+\log \frac{\sqrt{t}}{r})\|f(x){{\|}_{{{L}^{p}}({{\mathbb{R}}^{d}})}}, \tag{2.8}\end{equation} 其中$b_B=\frac1{|B|}\int_{B}b(y){\rm d}y,$ $ B$是以$x$为心半径$r_B<\sqrt{t}.$

证 对任意的的$x\in {\Bbb R}^d$和$t>0$,有 \begin{eqnarray*} &&\int_{{\Bbb R}^d}e^{-\frac{|x-y|^{2}}{t}}|(b(y)-b_B)f(y)|{\rm d}y\\ &=&\int_{\frac{|x-y|}{\sqrt{t}}\leq1}e^{-\frac{|x-y|^{2}}{t}}|(b(y)-b_B)f(y)|{\rm d}y+ \sum_{j=1}^\infty\int_{2^{j-1}<\frac{|x-y|}{\sqrt{t}}\leq2^{j}}e^{-\frac{|x-y|^{2}}{t}}|(b(y)-b_B)f(y)|{\rm d}y\\ &\leq& C\sum_{j=0}^\infty e^{-c'2^{2j}}\int_{\frac{|x-y|}{\sqrt{t}}\leq2^{j}}|(b(y)-b_B)f(y)|{\rm d}y\\ &\leq& C\sum_{j=0}^\infty e^{-c'2^{2j}}\bigg(\int_{B(x,2^{j}\sqrt{t})}|(b(y)-b_{B(x,2^{j}\sqrt{t})})f(y)|{\rm d}y \\ && +|b_B-b_{B(x,2^{j}\sqrt{t})}|\int_{B(x,2^{j}\sqrt{t})}|f(y)|{\rm d}y\bigg)\\ &\leq& C\|b\|_{BMO}t^{\frac d{2p'}}\sum_{j=0}^\infty 2^{\frac{jd}{p'}}e^{-c'2^{2j}}\|f\|_{L^p}+ C\sum_{j=0}^\infty e^{-c'2^{2j}}|b_B-b_{B(x,2^{j}\sqrt{t})}|\int_{B(x,2^{j}\sqrt{t})}|f(y)|{\rm d}y\\ &\leq& C\|b\|_{BMO}t^{\frac d2}\sum_{j=0}^\infty 2^{\frac{jd}{p'}}e^{-c'2^{2j}}\|f\|_{L^p} +C\|b\|_{BMO}t^{\frac d{2p'}}\sum_{j=0}^\infty j2^{\frac{jd}{p'}}e^{-c'2^{2j}}\|f\|_{L^p}\\ && +C\|b\|_{BMO}t^{\frac d{2p'}}\log\frac {\sqrt{t}}r\sum_{j=0}^\infty 2^{\frac{jd}{p'}}e^{-c'2^{2j}}\|f\|_{L^p}\\ &\leq& C\|b\|_{BMO}t^{\frac{d}{2p'}}\|f\|_{L^p}+C\|b\|_{BMO}t^{\frac d{2p'}}\log\frac {\sqrt{t}}r\|f\|_{L^p}\\ &\leq& C\|b\|_{BMO}t^{\frac{d}{2p'}}\bigg(1+\log\frac {\sqrt{t}}r\bigg)\|f\|_{L^p}.\end{eqnarray*}

定理1.1的证明 我们先证对任意的$f\in L^{p_{2}'}({\Bbb R}^d)$和$b\in \text{BMO}({{\mathbb{R}}^{d}})$,存在大于$0$的常数$C$,使得对任意的球$B\equiv B(x_{0},r),$ 当$r\geq\rho(x_{0})$时,有 \begin{equation}\label{eq:1.14} \frac{1}{|B|^{1/p_{1}}}\int_{B}|[b,I_{\alpha}^{L}]f(x)|{\rm d}x\leq C\|b\|_{BMO}\|f\|_{L^{p_{2}'}({\Bbb R}^d)}. \tag{2.9}\end{equation} 对任意的球$B(x_{0},r),$ 记 \begin{equation}\label{eq:1.15} [b,I_{\alpha}^{L}]f(x)=(b-b_{B})I_{\alpha}^{L}f(x)+I_{\alpha}^{L}((b-b_{B})f)(x), \tag{2.10}\end{equation} 其中$b_{B}=\frac{1}{|B|}\int_{B}|f(x)|{\rm d}x.$ 则我们只需证明 \begin{equation}\label{eq:1.16} \frac{1}{|B|^{1/p_{1}}}\int_{B}|(b(x)-b_{B})I_{\alpha}^{L}f(x)|{\rm d}x\leq C\|b\|_{BMO}\|f\|_{L^{p_{2}'}({\Bbb R}^d)} \tag{2.11}\end{equation} 和 \begin{equation}\label{eq:1.17} \frac{1}{|B|^{1/p_{1}}}\int_{B}|I_{\alpha}^{L}((b-b_{B})f)(x)|{\rm d}x\leq C\|b\|_{BMO}\|f\|_{L^{p_{2}'}({\Bbb R}^d)}.\tag{2.12}\end{equation} 利用Hölder不等式可得 \begin{eqnarray*} &&\frac{1}{|B|^{1/p_{1}}}\int_{B}|(b(x)-b_{B})I_{\alpha}^{L}f(x)|{\rm d}x\\ &\leq& C\|b\|_{BMO}|B|^{1/p_{2}-1/p_{1}} \bigg(\int_{B}|I_{\alpha}^{L}f(x)|^{p_{2}'}{\rm d}x\bigg)^{1/^{p_{2}'}}\\ &\leq& C\|b\|_{BMO}|B|^{1/p_{2}-1/p_{1}} \bigg(\int_{B}\bigg|\int_{0}^{r^{2}}\int_{{\Bbb R}^d}K_{t}^{L}(x,y)f(y){\rm d}yt^{\frac{\alpha}{2}}\frac{{\rm d}t}{t}\bigg|^{p_{2}'}{\rm d}x\bigg)^{1/^{p_{2}'}}\\ &&+C\|b\|_{BMO}|B|^{1/p_{2}-1/p_{1}} \bigg(\int_{B}\bigg|\int_{r^{2}}^{\infty}\int_{{\Bbb R}^d}K_{t}^{L}(x,y)f(y){\rm d}yt^{\frac{\alpha}{2}}\frac{{\rm d}t}{t}\bigg|^{p_{2}'}{\rm d}x\bigg)^{1/^{p_{2}'}}\\ &\triangleq& I_{1}+I_{2}.\end{eqnarray*} 下面我们分别对每一项进行估计. 首先来估计$I_{1}$,由(2.2)式和引理2.1, 结合极大算子$M$在$L^{p_{2}'}({\Bbb R}^d)$上的有界性,有 \begin{eqnarray*} I_{1}&\leq&C\|b\|_{BMO}|B|^{1/p_{2}-1/p_{1}} \bigg(\int_{B}\bigg|\int_{0}^{r^{2}}t^{\frac{\alpha-d}{2}} \int_{{\Bbb R}^d}e^{-\frac{|x-y|^{2}}{4t}}|f(y)|{\rm d}y\frac{{\rm d}t}{t} \bigg|^{p_{2}'}{\rm d}x\bigg)^{1/^{p_{2}'}}\\ &\leq&C\|b\|_{BMO}|B|^{1/p_{2}-1/p_{1}} \bigg(\int_{B}\bigg|\int_{0}^{r^{2}}t^{\frac{\alpha}{2}}\frac{{\rm d}t}{t} Mf(x)\bigg|^{p_{2}'}{\rm d}x\bigg)^{1/^{p_{2}'}}\\ &\leq&C\|b\|_{BMO}|B|^{1/p_{2}-1/p_{1}}r^{\alpha}\|Mf(x)\|_{L^{p_{2}'}({\Bbb R}^d)}\\ &\leq&C\|b\|_{BMO}\|f(x)\|_{L^{p_{2}'}({\Bbb R}^d)}.\end{eqnarray*} 现在我们来估计$I_{2}$. 由$N>\alpha$应用(2.4)式和引理2.2,注意到$\rho(x)\sim\rho(x_{0})\leq r,$ 则 \begin{eqnarray*} I_{2}&\leq&C\|b\|_{BMO}|B|^{1/p_{2}-1/p_{1}} \bigg(\int_{B}\bigg|\int_{r^{2}}^{\infty} \bigg(\frac{\rho(x)}{\sqrt{t}}\bigg)^{N}t^{\frac{\alpha-d}{2}} \int_{{\Bbb R}^d}e^{-\frac{|x-y|^{2}}{4t}}|f(y)|{\rm d}y\frac{{\rm d}t}{t}\bigg|^{p_{2}'}{\rm d}x\bigg)^{1/^{p_{2}'}}\\ &\leq&C\|b\|_{BMO}|B|^{1-1/p_{1}}[\rho(x_{0})]^{N}\int_{r^{2}}^{\infty}t^{\frac{\alpha-d-N}{2}}t^{\frac{d}{2p_{2}}}\frac{{\rm d}t}{t} \|f(x)\|_{L^{p_{2}'}({\Bbb R}^d)}\\ &\leq&C\|b\|_{BMO}\|f(x)\|_{L^{p_{2}'}({\Bbb R}^d)}.\end{eqnarray*} 因此(2.11)式得证. 下证(2.12)式,对任意的$x\in B$,令 \begin{eqnarray*} II&=&\frac{1}{|B|^{1/p_{1}}}\int_{B}|I_{\alpha}^{L}((b-b_{B})f)(x)|{\rm d}x\\ &\leq&C\frac{1}{|B|^{1/p_{1}}}\int_{B}\bigg|\int_{0}^{\infty}\int_{{\Bbb R}^d}K_{t}^ {L}(x,y)(b-b_{B})f(y){\rm d}yt^{\frac{\alpha}{2}}\frac{{\rm d}t}{t}\bigg|{\rm d}x\\ &\leq&C\frac{1}{|B|^{1/p_{1}}}\int_{B}\bigg|\int_{0}^{r^{2}}\int_{{\Bbb R}^d}K_{t}^ {L}(x,y)(b-b_{B})f(y){\rm d}yt^{\frac{\alpha}{2}}\frac{{\rm d}t}{t}\bigg|{\rm d}x\\ &&+C\frac{1}{|B|^{1/p_{1}}}\int_{B}\bigg|\int_{r^{2}}^{\infty}\int_{{\Bbb R}^d}K_{t} ^{L}(x,y)(b-b_{B})f(y){\rm d}yt^{\frac{\alpha}{2}}\frac{{\rm d}t}{t}\bigg|{\rm d}x\\ &\triangleq&II_{1}+II_{2}.\end{eqnarray*} 由(2.2)式和引理2.3,有 \begin{eqnarray*} II_{1}&\leq& C\frac{1}{|B|^{1/p_{1}}}\int_{B}\bigg|\int_{0}^{r^{2}}t^{\frac{\alpha-d}{2}} \int_{{\Bbb R}^d}e^{-\frac{|x-y|^{2}}{4t}}(b(y)-b_{B})f(y){\rm d}y\frac{{\rm d}t}{t}\bigg|{\rm d}x\\ &\leq&C\|b\|_{BMO}|B|^{1-1/p_{1}}\int_{0}^{r^{2}}t^{\frac{\alpha-d}{2}}t^{\frac{d}{2p_{2}}}\frac{{\rm d}t}{t} \|f(x)\|_{L^{p_{2}'}({\Bbb R}^d)}\\ &\leq&C\|b\|_{BMO}\|f(x)\|_{L^{p_{2}'}({\Bbb R}^d)}.\end{eqnarray*} 为证$II_{2}$,由$N>\alpha$,结合(2.4)式和引理2.4及$\rho(x)\sim\rho(x_{0})\leq r,$ 则 \begin{eqnarray*} II_{2}&\leq& C\frac{1}{|B|^{1/p_{1}}}\int_{B}\bigg|\int_{r^{2}}^{\infty}\bigg(\frac{\rho(x)}{\sqrt{t}}\bigg)^{N} t^{\frac{\alpha-d}{2}}\int_{{\Bbb R}^d}e^{-\frac{|x-y|^{2}}{4t}}|(b(y)-b_{B})f(y) |{\rm d}y\frac{{\rm d}t}{t}\bigg|{\rm d}x\\ &\leq&C\|b\|_{BMO}|B|^{1-1/p_{1}}[\rho(x_{0})]^{N}\int_{r^{2}}^{\infty}t^{\frac{\alpha-d-N}{2}}t^{\frac{d}{2p_{2}}}\frac{{\rm d}t}{t} \|f(x)\|_{L^{p_{2}'}({\Bbb R}^d)}\\ &\leq&C\|b\|_{BMO}\|f(x)\|_{L^{p_{2}'}({\Bbb R}^d)}.\end{eqnarray*} 因此(2.12)式得证. 结合(2.11)和(2.12)式可得,当$f\in L^{p_{2}'}({\Bbb R}^d)$时,对任意的$x\in {\Bbb R}^d$, $[b,I_{\alpha}^{L}]f(x)$是几乎处处有限的.

要完成定理1.1的证明,我们还需证明: 当$0<r<\rho(x_{0})$和$f\in L^{p_{2}'}({\Bbb R}^d)$,有 \begin{equation}\label{eq:1.18} \frac{1}{|B|^{1/p_{1}}}\int_{B}|[b,I_{\alpha}^{L}]f(x)-([b,I_{\alpha}^{L}]f)_{B}|{\rm d}x\leq C\|b\|_{BMO}\|f\|_{L^{p_{2}'}({\Bbb R}^d)}. \tag{2.13}\end{equation} 注意到 \[[b,I_{\alpha}^{L}]f=(b-b_{B})I_{\alpha}^{L}f+I_{\alpha}^{L}(b-b_{B})f(x)\triangleq A_{1}f+A_{2}f,\] 则 \begin{eqnarray*} &&\frac{1}{|B|^{1/p_{1}}}\int_{B}|[b,I_{\alpha}^{L}]f(x)-([b,I_{\alpha}^{L}]f)_{B}|{\rm d}x\\ &\leq&\frac{1}{|B|^{1/p_{1}}}\int_{B}|A_{1}f(x)-(A_{1}f)_{B}|{\rm d}x+\frac{1}{|B|^{1/p_{1}}}\int_{B}|A_{2}f(x)-(A_{2}f)_{B}|{\rm d}x\\ &\triangleq& J_{1}+J_{2}.\end{eqnarray*} 取一个仅与$I_{\alpha}^{L}f,b$和$B$相关的常数$C_{B}$,且令 \[C_{B}=\int_{r^{2}}^{\infty}\int_{{\Bbb R}^d}K_{t}^{L}(x_{0},y)(b(y)-b_{B})f(y){\rm d}yt^{\frac{\alpha}{2}}\frac{{\rm d}t}{t},\] 则 \begin{eqnarray*} J_{2}&\leq&C\frac{1}{|B|^{1/p_{1}}}\int_{B}|A_{2}f(x)-C_{B}|{\rm d}x\\ &\leq&C\frac{1}{|B|^{1/p_{1}}}\int_{B}\bigg|\int_{0}^{\infty}\int_{{\Bbb R}^d} K_{t}^{L}(x,y)(b(y)-b_{B})f(y){\rm d}yt^{\frac{\alpha}{2}}\frac{{\rm d}t}{t}-C_{B}\bigg|{\rm d}x\\ &\leq&C\frac{1}{|B|^{1/p_{1}}}\int_{B}\bigg|\int_{0}^{r^{2}}\int_{{\Bbb R}^d}K_{t}^ {L}(x,y)(b(y)-b_{B})f(y){\rm d}yt^{\frac{\alpha}{2}}\frac{{\rm d}t}{t}\bigg|{\rm d}x\\ &&+C\frac{1}{|B|^{1/p_{1}}}\int_{B}\bigg|\int_{r^{2}}^{\infty}\int_{{\Bbb R}^d}[K_{t}^ {L}(x,y)-K_{t}^{L}(x_{0},y)](b(y)-b_{B})f(y){\rm d}yt^{\frac{\alpha}{2}}\frac{{\rm d}t}{t}\bigg|{\rm d}x\\ &\triangleq&J_{21}+J_{22}.\end{eqnarray*} 类似于$II_{1}$的证明,易证$J_{21}\leq C\|b\|_{BMO}\|f(x)\|_{L^{p_{2}'}({\Bbb R}^d)}$.现在我们来估计$J_{22}$. 注意到对任意的$x\in B,|x-x_{0}|<r\leq\sqrt{t}$和$\delta>d(1/p_{1}-1)$,由(2.5)式和引理2.3,有 \begin{eqnarray*} J_{22}&\leq&C\frac{1}{|B|^{1/p_{1}}}\int_{B}\bigg|\int_{r^{2}}^{\infty}\int_{{\Bbb R}^d} [K_{t}^{L}(x,y)-K_{t}^{L}(x_{0},y)](b(y)-b_{B})f(y){\rm d}yt^{\frac{\alpha}{2}}\frac{{\rm d}t}{t} \bigg|{\rm d}x\\ &\leq&C\frac{1}{|B|^{1/p_{1}}}\int_{B}\bigg|\int_{r^{2}}^{\infty} \bigg(\frac{|x-x_{0}|}{\sqrt{t}}\bigg)^{\delta}t^{\frac{\alpha-d}{2}} \int_{{\Bbb R}^d}e^{-A\frac{|x-y|^{2}}{t}}|(b(y)-b_{B})f(y)|{\rm d}y\frac{{\rm d}t}{t}\bigg|{\rm d}x\\ &\leq&C\|b\|_{BMO}|B|^{1-1/p_{1}}\int_{r^{2}}^{\infty} \bigg(\frac{r}{\sqrt{t}}\bigg)^{\delta}\log\frac{\sqrt{t}}r t^{\frac{\alpha-d}{2}} t^{\frac{d}{2p_{2}}}\frac{{\rm d}t}{t} \|f(x)\|_{L^{p_{2}'}({\Bbb R}^d)}\\ &\leq&C\|b\|_{BMO}r^{d-d/p_{1}}r^{\alpha-d}r^{\frac{d}{p_{2}}}\|f(x)\|_{L^{p_{2}'}({\Bbb R}^d)}\\ &\leq&C\|b\|_{BMO}\|f(x)\|_{L^{p_{2}'}({\Bbb R}^d)}.\end{eqnarray*} 由此证得$J_{2}\leq C\|b\|_{BMO}\|f(x)\|_{L^{p_{2}'}({\Bbb R}^d)}$. 现在我们来估计$J_{1}$,注意到 \begin{eqnarray*} J_{1}&\leq&C\frac{1}{|B|^{1/p_{1}}}\int_{B}|A_{1}f(x)|{\rm d}x\\ &=&C\frac{1}{|B|^{1/p_{1}}}\int_{B}|(b(x)-b_{B})I_{\alpha}^{L}f(x)|{\rm d}x\\ &\leq&C\|b\|_{BMO}|B|^{1/p_{2}-1/p_{1}} \bigg(\int_{B}|I_{\alpha}^{L}f(x)|^{p_{2}'}{\rm d}x\bigg)^{1/^{p_{2}'}}\\ &\leq&C\|b\|_{BMO}|B|^{1/p_{2}-1/p_{1}}\bigg\{ \bigg(\int_{B}\bigg|\int_{0}^{r^{2}}\int_{{\Bbb R}^d}K_{t}^{L}(x,y)f(y){\rm d}yt^{\frac{\alpha}{2}} \frac{{\rm d}t}{t}\bigg|^{p_{2}'}{\rm d}x\bigg)^{1/^{p_{2}'}}\\ &&+\bigg(\int_{B}\bigg|\int_{r^{2}}^{\infty}\int_{\mathbb{R}^d}K_{t}^{L} (x,y)f(y){\rm d}yt^{\frac{\alpha}{2}}\frac{{\rm d}t}{t}\bigg|^{p_{2}'}{\rm d}x\bigg)^{1/^{p_{2}'}}\bigg\}\\ &\triangleq&J_{11}+J_{12}.\end{eqnarray*} 类似于$I_{1}$的估计,易证$J_{11}\leq C\|b\|_{BMO}\|f(x)\|_{L^{p_{2}'}({\Bbb R}^d)}.$

现在估计$J_{12}$,记$J_{12}=J_{121}+J_{122}$. 对于$J_{121}$,由(2.2)式,有 \begin{eqnarray*} J_{121}&=&C\|b\|_{BMO}|B|^{1/p_{2}-1/p_{1}} \bigg(\int_{B}\bigg|\int_{r^{2}}^{\infty}\int_{{\Bbb R}^d} K_{t}^{L}(x_{0},y)f(y){\rm d}yt^{\frac{\alpha}{2}}\frac{{\rm d}t}{t}\bigg|^{p_{2}'}{\rm d}x\bigg)^{1/^{p_{2}'}}\\ &\leq&C\|b\|_{BMO}|B|^{1/p_{2}-1/p_{1}} \bigg(\int_{B}\bigg|\int_{r^{2}}^{\infty}t^{\frac{\alpha-d}{2}} \int_{{\Bbb R}^d}e^{-\frac{|x_{0}-y|^{2}}{4t}}f(y){\rm d}y\frac{{\rm d}t}{t}\bigg|^{p_{2}'}{\rm d}x\bigg)^{1/^{p_{2}'}}\\ &\leq&C\|b\|_{BMO}|B|^{1/p_{2}-1/p_{1}}r^{\alpha-d+d/p_{2}}|B|^{d/p_{2}'} \|f(x)\|_{L^{p_{2}'}({\Bbb R}^d)}\\ &\leq&C\|b\|_{BMO}\|f(x)\|_{L^{p_{2}'}({\Bbb R}^d)}.\end{eqnarray*} 对任意的$x\in B$,当$|x-x_{0}|<r\leq\sqrt{t}$和$\delta>d(1/p_{1}-1)$时,由(2.5)式和引理2.3,有 \begin{eqnarray*} J_{122}&=&C\|b\|_{BMO}|B|^{1/p_{2}-1/p_{1}}\bigg(\int_{B}\bigg|\int_{r^{2}}^{\infty}\int_{{\Bbb R}^d} [K_{t}^{L}(x,y)-K_{t}^{L}(x_{0},y)]f(y){\rm d}yt^{\frac{\alpha}{2}}\frac{{\rm d}t}{t} \bigg|^{p_{2}'}{\rm d}x\bigg)^{1/^{p_{2}'}}\\ &\leq&C\|b\|_{BMO}|B|^{1/p_{2}-1/p_{1}} \bigg(\int_{B}\bigg|\int_{r^{2}}^{\infty}\bigg(\frac{|x-x_{0}|}{\sqrt{t}} \bigg)^{\delta}t^{\frac{\alpha-d}{2}} \int_{{\Bbb R}^d}e^{-A\frac{|x-y|^{2}}{4t}} |f(y)|{\rm d}y\frac{{\rm d}t}{t}\bigg|^{p_{2}'}{\rm d}x\bigg)^{1/^{p_{2}'}}\\ &\leq&C\|b\|_{BMO}|B|^{1/p_{2}-1/p_{1}}|B|^{1/p_{2}'}\int_{r^{2}}^{\infty} \bigg(\frac{r}{\sqrt{t}}\bigg)^ {\delta}\log\frac{\sqrt{t}}rt^{\frac{\alpha-d}{2}}t^{d/2p_{2}}\frac{{\rm d}t}{t} \|f(x)\|_{L^{p_{2}'}({\Bbb R}^d)}\\ &\leq&C\|b\|_{BMO}r^{d-d/p_{1}}r^{\alpha-d}r^{d/p_{2}}\|f(x)\|_{L^{p_{2}'}({\Bbb R}^d)}\\ &\leq&C\|b\|_{BMO}\|f(x)\|_{L^{p_{2}'}({\Bbb R}^d)}.\end{eqnarray*} 由此完成了(2.13)式的证明,即定理1.1得证.

下面我们来证明定理1.2. 在证明定理1.2之前我们先介绍几个证明中需要用到的引理.

引理3.1^[4] 设$0<p<1$,$f\in\Lambda_{L}^{1/p-1}({\Bbb R}^d)$, 则对任意的$x\in {\Bbb R}^d$和$t>0$,有 \begin{equation}\label{eq:1.19}\int_{{\Bbb R}^d}e^{-\frac{|x-y|^{2}}{t}}|f(y)|{\rm d}y\leq Ct^{\frac{d}{2p}}\|f\|_{\Lambda_{L}^{1/p-1}({\Bbb R}^d)}.\tag{3.1}\end{equation}

引理3.2  设$0<p<1$,$f\in\Lambda_{L}^{1/p-1}({\Bbb R}^d)$,当$b\in BMO({\Bbb R}^d)$时,存在常数$\lambda$,使得对任意的$x\in {\Bbb R}^d$和$t$,有 \begin{equation}\label{eq:1.20} \int_{{\Bbb R}^d}e^{-\frac{|x-y|^{2}}{t}}|(b(y)-\lambda)f(y)|{\rm d}y\leq C\|b\|_{BMO}t^{\frac{d}{p}}\bigg(1+\log\frac {\sqrt{t}}r\bigg)\|f\|_{\Lambda_L^{1/p-1}}, \tag{3.2}\end{equation} 其中$b_B=\frac1{|B|}\int_{B}b(y){\rm d}y,$ $ B$是以$x$为心半径$r_B<\sqrt{t}.$

证 对任意的的$x\in {\Bbb R}^d$和$t>0$,有 \begin{eqnarray*} &&\int_{{\Bbb R}^d}e^{-\frac{|x-y|^{2}}{t}}|(b(y)-b_B)f(y)|{\rm d}y \\ &=&\int_{\frac{|x-y|}{\sqrt{t}}\leq1}e^{-\frac{|x-y|^{2}}{t}}|(b(y)-b_B)f(y)|{\rm d}y+ \sum_{j=1}^\infty\int_{2^{j-1}<\frac{|x-y|}{\sqrt{t}}\leq2^{j}}e^{-\frac{|x-y|^{2}}{t}}|(b(y)-b_B)f(y)|{\rm d}y\\ &\leq& C\sum_{j=0}^\infty e^{-c'2^{2j}}\int_{\frac{|x-y|}{\sqrt{t}}\leq2^{j}}|(b(y)-b_B)f(y)|{\rm d}y\\ &\leq& C\sum_{j=0}^\infty e^{-c'2^{2j}}\bigg(\int_{B(x,2^{j}\sqrt{t})}|(b(y)-b_{B(x,2^{j}\sqrt{t})})f(y)|{\rm d}y \\ && +|b_B-b_{B(x,2^{j}\sqrt{t})}|\int_{B(x,2^{j}\sqrt{t})}|f(y)|{\rm d}y\bigg)\\ &\leq& C\|b\|_{BMO}t^{\frac d{p}}\sum_{j=0}^\infty 2^{\frac{jd}{p}}e^{-c'2^{2j}}\bigg(\frac{1}{|B|^{1+2(\frac1p-1)}}\int_{B(x,2^{j}\sqrt{t})}|f(y)|^2dy\bigg)^{1/2}\\ &&+C\sum_{j=0}^\infty e^{-c'2^{2j}}|b_B-b_{B(x,2^{j}\sqrt{t})}|\int_{B(x,2^{j}\sqrt{t})}|f(y)|{\rm d}y\\ &\leq& C\|b\|_{BMO}t^{\frac d2}\sum_{j=0}^\infty 2^{\frac{jd}{p}}e^{-c'2^{2j}}\|f\|_{\Lambda_L^{1/p-1}} +C\|b\|_{BMO}t^{\frac d{p}}\sum_{j=0}^\infty j2^{\frac{jd}{p}}e^{-c'2^{2j}}\|f\|_{\Lambda_L^{1/p-1}}\\ &\quad&+C\|b\|_{BMO}t^{\frac d{p}}\log\frac {\sqrt{t}}r\sum_{j=0}^\infty 2^{\frac{jd}{p}}e^{-c'2^{2j}}\|f\|_{\Lambda_L^{1/p-1}}\\ &\leq& C\|b\|_{BMO}t^{\frac{d}{p}}\|f\|_{L^p}+C\|b\|_{BMO}t^{\frac d{p}}\log\frac {\sqrt{t}}r\|f\|_{\Lambda_L^{1/p-1}}\\ &\leq& C\|b\|_{BMO}t^{\frac{d}{p}}\bigg(1+\log\frac {\sqrt{t}}r\bigg)\|f\|_{\Lambda_L^{1/p-1}}.\end{eqnarray*} 证毕.

引理3.3^[1] 设$0<p<1$,$f\in\Lambda_{L}^{1/p-1}({\Bbb R}^d)$和$x_{0}\in {\Bbb R}^d$,对球$B\equiv B(x_{0},r)$,当$r<\rho(x_{0})$时,则 \[\frac{1}{|B|^{1/p}}\int_{B}|f(y)|{\rm d}y\leq C\rho(x_{0})^{d(1/p-1)}\|f\|_{\Lambda_{L}^{1/p-1}({\Bbb R}^d)}.\]

下面我们来证明定理1.2. 先证对任意的球$B=B(x_{0},r)$和$\rho(x_{0})\leq r$,当$\rho(x_{0})\leq r$且$f\in\Lambda_{L}^{1/p_{2}-1}({\Bbb R}^d)$时,存在常数$C$使得 \begin{equation}\label{eq:1.21} \frac{1}{|B|^{1/p_{1}}}\int_{B}|[b,I_{\alpha}^{L}]f(x)|{\rm d}x\leq C\|b\|_{BMO}\|f\|_{\Lambda_{L}^{1/p_{2}-1}({\Bbb R}^d)}.\tag{3.3}\end{equation} 对任意的球$B(x_{0},r)$记 \[[b,I_{\alpha}^{L}]f=(b-b_{B})I_{\alpha}^{L}f+I_{\alpha}^{L}(b-b_{B})f,\] 则只需证 \begin{equation}\label{eq:1.22} \frac{1}{|B|^{1/p_{1}}}\int_{B}|(b(x)-b_{B})I_{\alpha}^{L}f(x)|{\rm d}x\leq C\|b\|_{BMO}\|f\|_{\Lambda_{L}^{1/p_{2}-1}({\Bbb R}^d)} \tag{3.4}\end{equation} 和 \begin{equation}\label{eq:1.23} \frac{1}{|B|^{1/p_{1}}}\int_{B}|I_{\alpha}^{L}((b-b_{B})f)(x)|{\rm d}x\leq C\|b\|_{BMO}\|f\|_{\Lambda_{L}^{1/p_{2}-1}({\Bbb R}^d)}.\tag{3.5}\end{equation} 对任意的$x\in B(x_{0},r)$记 \begin{eqnarray*} I_{\alpha}^{L}f(x) &=& \int_{0}^{r^{2}}\int_{{\Bbb R}^d}K_{t}^{L}(x,y)f(y){\rm d}yt^{\frac{\alpha}{2}}\frac{{\rm d}t}{t} +\int_{r^{2}}^{\infty}\int_{{\Bbb R}^d}K_{t}^{L}(x,y)f(y){\rm d}yt^{\frac{\alpha}{2}}\frac{{\rm d}t}{t}\\ &\triangleq&I_{1}(f)(x)+I_{2}(f)(x).\end{eqnarray*} 设$f=f\chi_{(4B)}+f\chi_{(4B)^{c}},$ 由(2.2)式和引理2.1, 结合Hölder不等式及Hardy-Littlewood极大算子$M$在$L^{s}({\Bbb R}^d)$ ($1<s<\infty$)上的有界性,则 \begin{eqnarray*} &&\frac{1}{|B|^{1/p_{1}}}\int_{B}|(b(x)-b_{B})I_{1}(f\chi_{(4B)})(x)|{\rm d}x\\ &\leq& C\frac{1}{|B|^{1/p_{1}}} \bigg(\int_{B}|b(x)-b_{B}|^{s'}{\rm d}x\bigg)^{1/s'} \bigg(\int_{B}|I_{1}(f\chi_{(4B)})(x)|^{s}{\rm d}x\bigg)^{1/s}\\ &\leq& C\|b\|_{BMO}|B|^{1/s'-1/p_{1}} \bigg(\int_{B}\bigg|\int_{0}^{r^{2}}t^{\frac{\alpha-d}{2}} \int_{{\Bbb R}^d}e^{-\frac{|x-y|^{2}}{4t}}f(y)\chi_{(4B)^{c}}(y){\rm d}y \frac{{\rm d}t}{t}\bigg|^{s}{\rm d}x\bigg)^{1/s}\\ &\leq& C\|b\|_{BMO}|B|^{1/s'-1/p_{1}}\bigg(\int_{B}\bigg|M(f\chi_{(4B)})(x)\int_{0}^{r^{2}}t^{\frac{\alpha}{2}} \frac{{\rm d}t}{t}\bigg|^{s}{\rm d}x\bigg)^{1/s}\\ &\leq& C\|b\|_{BMO}|B|^{1/s'-1/p_{1}}r^{\alpha} \bigg(\int_{B}[M(f\chi_{(4B)})(x)]^{s}{\rm d}x\bigg)^{1/s}\\ &\leq& C\|b\|_{BMO}|B|^{1/s'-1/p_{1}}r^{\alpha}|B|^{1/p_{2}-1+1/s}|B|^{-(1/p_{2}-1)} \bigg(\frac{1}{|4B|}\int_{B}|f(x)|^{s}{\rm d}x\bigg)^{1/s}\\ &\leq& C\|b\|_{BMO}\|f\|_{\Lambda_{L}^{1/p_{2}-1}({\Bbb R}^d)}.\end{eqnarray*} 注意到当$x\in B,y\in(4B)^{c}$时,$|x-y|>3r$. 且设$N'>d/p_{2}$,由(2.2)式可得 \begin{eqnarray*} |I_{1}(f\chi_{(4B)^{c}})(x)| &\leq& C\int_{0}^{r^{2}}t^{\frac{\alpha-d}{2}} \int_{{\Bbb R}^d}e^{-\frac{|x-y|^{2}}{4t}}f(y)\chi_{(4B)^{c}}(y){\rm d}y\frac{{\rm d}t}{t}\\ &\leq&C\int_{0}^{r^{2}}t^{\frac{\alpha-d}{2}}\sum_{j=1}^\infty\int_{2^{j}r<|x-y|\leq2^{j+1}r} e^{-\frac{|x-y|^{2}}{4t}}|f(y)|{\rm d}y\frac{{\rm d}t}{t}\\ &\leq&C\int_{0}^{r^{2}}t^{\frac{\alpha-d}{2}}\sum_{j=1}^\infty e^{-\frac{A2^{jr}r^{2}}{t}}\int_{B(x,2^{j+1}r)}|f(y)|{\rm d}y\frac{{\rm d}t}{t}\\ &\leq&C\int_{0}^{r^{2}}t^{\frac{\alpha-d}{2}}\sum_{j=1}^\infty \bigg(\frac{2^{j}r}{\sqrt{t}}\bigg)^{-N'}\frac{2^{jd/p_{2}}r^{d/p_{2}}}{|B(x,2^{j+1}r)|^{d/p_{2}}}\int_{B(x,2^{j+1}r)}|f(y)|{\rm d}y\frac{{\rm d}t}{t}\\ &\leq&Cr^{-N'+d/p_{2}}\int_{0}^{r^{2}}t^{\frac{\alpha-d+N'}{2}}\frac{{\rm d}t}{t} \sum_{j=1}^\infty2^{-j(N'-d/p_{2})}\|f\|_{\Lambda_{L}^{1/p_{2}-1}({\Bbb R}^d)}\\ &\leq&Cr^{d(p_{1}-1)}\|f\|_{\Lambda_{L}^{1/p_{2}-1}({\Bbb R}^d)}.\end{eqnarray*} 则 \begin{eqnarray*} &&\frac{1}{|B|^{1/p_{1}}}\int_{B}|(b(x)-b_{B})I_{1}(f\chi_{(4B)^{c}})(x)|{\rm d}x \\ &\leq& C\|b\|_{BMO}|B|^{1/s'-1/p_{1}}\|I_{1}(f\chi_{(4B)^{c}})(x)\|_{L^{^{s}}({\Bbb R}^d)}\\ &\leq& C\|b\|_{BMO}|B|^{1/s'-1/p_{1}}r^{d(1/p_{1}-1)+\frac{d}{s}}\|f\|_{\Lambda_{L}^{1/p_{2}-1}({\Bbb R}^d)}\\ &\leq& C\|b\|_{BMO}\|f\|_{\Lambda_{L}^{1/p_{2}-1}({\Bbb R}^d)}.\end{eqnarray*} 即得 \[\frac{1}{|B|^{1/p_{1}}}\int_{B}|(b(x)-b_{B})I_{1}(f)(x)|{\rm d}x\leq C\|b\|_{BMO}\|f\|_{\Lambda_{L}^{1/p_{2}-1}({\Bbb R}^d)}.\] 类似于定理1.1中$II_{2}$的估计方法,由Hölder和(2.4)式知当$N>\alpha$结合引理3.1可得 \[\frac{1}{|B|^{1/p_{1}}}\int_{B}|(b(x)-b_{B})I_{2}(f)(x)|{\rm d}x\leq C\|b\|_{BMO}\|f\|_{\Lambda_{L}^{1/p_{2}-1}({\Bbb R}^d)}.\] 则(3.4)式得证. 下证(3.5)式. 令 \begin{eqnarray*} II&=&\frac{1}{|B|^{1/p_{1}}}\int_{B}|I_{\alpha}^{L}((b-b_{B})f)(x)|{\rm d}x\\ &\leq&C\frac{1}{|B|^{1/p_{1}}}\int_{B}\bigg|\int_{0}^{\infty}\int_{{\Bbb R}^d}K_{t}^ {L}(x,y)(b-b_{B})f(y){\rm d}yt^{\frac{\alpha}{2}}\frac{{\rm d}t}{t}\bigg|{\rm d}x\\ &\leq&C\frac{1}{|B|^{1/p_{1}}}\int_{B}\bigg|\int_{0}^{r^{2}}\int_{{\Bbb R}^d}K_{t}^ {L}(x,y)(b-b_{B})f(y){\rm d}yt^{\frac{\alpha}{2}}\frac{{\rm d}t}{t}\bigg|{\rm d}x\\ &&+C\frac{1}{|B|^{1/p_{1}}}\int_{B}\bigg|\int_{r^{2}}^{\infty}\int_{{\Bbb R}^d}K_{t} ^{L}(x,y)(b-b_{B})f(y){\rm d}yt^{\frac{\alpha}{2}}\frac{{\rm d}t}{t}\bigg|{\rm d}x\\ &\triangleq&II_{1}+II_{2}.\end{eqnarray*} 由引理3.2和定理1.1中对$II_{1}$的估计得 \[II_{1}\leq C\|b\|_{BMO}\|f\|_{\Lambda_{L}^{1/p_{2}-1}({\Bbb R}^d)}.\] 至于$II_{2}$,当$N>\alpha$时,由(2.4)式和引理3.2及定理1.1中对$I_{2}$的估计方法可得 \[II_{2}\leq C\|b\|_{BMO}\|f\|_{\Lambda_{L}^{1/p_{2}-1}({\Bbb R}^d)}.\] 即(3.5)式得证. 由(3.4)和(3.5)式即证得对任意的$x\in {\Bbb R}^d$当$f\in \Lambda_{L}^{1/p_{2}-1}({\Bbb R}^d)$时,$[b,I_{\alpha}^{L}]f$是几乎处处有限的.

要完成定理1.2的证明,我们还需找到一个大于0的常数$C,$ 使得对任意的球$B(x_{0},r)$当$0<r<\rho(x_{0})$和$f\in\Lambda_{L}^{1/p_{2}-1}({\Bbb R}^d)$,有 \begin{equation}\label{eq:1.24} \frac{1}{|B|^{1/p_{1}}}\int_{B}|[b,I_{\alpha}^{L}]f(x)-([b,I_{\alpha}^{L}]f)_{B}|{\rm d}x\leq C\|b\|_{BMO}\|f\|_{\Lambda_{L}^{1/p_{2}-1}({\Bbb R}^d)}.\tag{3.6}\end{equation} 注意到 \[[b,I_{\alpha}^{L}]f=(b-b_{B})I_{\alpha}^{L}f+I_{\alpha}^{L}(b-b_{B})f(x)\triangleq A_{1}f+A_{2}f,\] 则有 \begin{eqnarray*} &&\frac{1}{|B|^{1/p_{1}}}\int_{B}|[b,I_{\alpha}^{L}]f(x)-([b,I_{\alpha}^{L}]f)_{B}|{\rm d}x\\ &\leq&\frac{1}{|B|^{1/p_{1}}}\int_{B}|A_{1}f(x)-(A_{1}f)_{B}|{\rm d}x+\frac{1}{|B|^{1/p_{1}}}\int_{B}|A_{2}f(x)-(A_{2}f)_{B}|{\rm d}x\\ &\triangleq& J_{1}+J_{2}.\end{eqnarray*} 现在我们分别来估计$J_{1}$和$J_{2}$. 先估计$J_{2}$,令 \[C_{B}=\int_{r^{2}}^{\infty}\int_{{\Bbb R}^d}K_{t}^{L}(x_{0},y)(b(y)-b_{B})f(y){\rm d}yt^{\frac{\alpha}{2}}\frac{{\rm d}t}{t},\] 则有 \begin{eqnarray*} J_{2}&\leq&\frac{C}{|B|^{1/p_{1}}}\int_{B}|A_{2}f(x)-C_{B}|{\rm d}x\\ &\leq&\frac{C}{|B|^{1/p_{1}}}\int_{B}\bigg|\int_{0}^{\infty}\int_{{\Bbb R}^d} K_{t}^{L}(x,y)(b(y)-b_{B})f(y){\rm d}yt^{\frac{\alpha}{2}}\frac{{\rm d}t}{t}-C_{B}\bigg|{\rm d}x\\ &\leq&\frac{C}{|B|^{1/p_{1}}}\int_{B}\bigg|\int_{0}^{r^{2}}\int_{{\Bbb R}^d}K_{t}^ {L}(x,y)(b(y)-b_{B})f(y){\rm d}yt^{\frac{\alpha}{2}}\frac{{\rm d}t}{t}\bigg|{\rm d}x\\ &&+\frac{C}{|B|^{1/p_{1}}}\int_{B}\bigg|\int_{r^{2}}^{\infty}\int_{{\Bbb R}^d}[K_{t}^ {L}(x,y)-K_{t}^{L}(x_{0},y)](b(y)-b_{B})f(y){\rm d}yt^{\frac{\alpha}{2}}\frac{{\rm d}t}{t}\bigg|{\rm d}x\\ &\triangleq&J_{21}+J_{22}.\end{eqnarray*} 现在我们来估计$J_{22}(f)$,注意到对任意的$x\in B$,$|x-x_{0}|<r\leq\sqrt{t}$和$\delta>d(1/p_{1}-1)$,应用(2.5)式和引理3.2,有 \[J_{22}(f)\leq C\|b\|_{BMO}\|f\|_{L^{p_{2}'}({\Bbb R}^d)}. \] 记 \[f=(f-f_{B})\chi_{(4B)}+(f-f_{B})\chi_{(4B)^{c}}+f_{B}.\] 由(2.2)式和引理3.2,结合极大算子$M_{s}$在$L^{p_{2}'}({\Bbb R}^d)$的有界性,同时$\Lambda_{L}^{1/p_{2}-1}({\Bbb R}^d)\subseteq \Lambda^{1/p_{2}-1}({\Bbb R}^d),$ 有 \begin{eqnarray*} J_{21}&\leq& C\frac{1}{|B|^{1/p_{1}}}\int_{B}\int_{0}^{r^{2}}t^{\frac{\alpha-d}{2}} \int_{{\Bbb R}^d}e^{-\frac{|x-y|^{2}}{4t}}|b(y)-b_{B}||(f(y)-f_{B})\chi_{(4B)}(y)|{\rm d}y\frac{{\rm d}t}{t}{\rm d}x\\ &&+ C\frac{1}{|B|^{1/p_{1}}}\int_{B}\int_{0}^{r^{2}}t^{\frac{\alpha-d}{2}} \int_{{\Bbb R}^d}e^{-\frac{|x-y|^{2}}{4t}}|b(y)-b_{B}||(f(y)-f_{B})\chi_{(4B)^c}(y)|{\rm d}y\frac{{\rm d}t}{t}{\rm d}x\\ &&+C\frac{1}{|B|^{1/p_{1}}}\int_{B}\int_{0}^{r^{2}}t^{\frac{\alpha-d}{2}} \int_{{\Bbb R}^d}e^{-\frac{|x-y|^{2}}{4t}}|b(y)-b_{B}||f_{B}|{\rm d}y\frac{{\rm d}t}{t}{\rm d}x\\ &=&J_{21}^1+J_{21}^2+J_{21}^3.\end{eqnarray*} 现在来估计$J_{21}^1.$ 取$r>s,$利用Hardy-Littlewood极大函数在$L^{r/s}$上的有界性可得 \begin{eqnarray*} J_{21}^1&\leq&C\|b\|_{BMO}r^{\alpha}|B|^{\frac{1}{2}-1/p_{1}} \bigg\{\int_{B}[M(|(f-f_{B})\chi_{(4B)}|^s)(x)]^{r/s}{\rm d}x\bigg\}^{\frac{1}{r}}\\ &\leq&C\|b\|_{BMO}r^{\alpha}|B|^{\frac{1}{2}-1/p_{1}}|B|^{1/p_{2} -\frac{1}{2}}|4B|^{-(1/p_{2}-1)} \bigg\{\frac{1}{|4B|}\int_{4B}|f(x)-f_{B}|^{s}{\rm d}x\bigg\}^{\frac{1}{s}}\\ &\leq&C\|b\|_{BMO}\|f(x)\|_{\Lambda^{1/p_{2}-1}({\Bbb R}^d)}\\ &\leq&C\|b\|_{BMO}\|f(x)\|_{\Lambda_{L}^{1/p_{2}-1}({\Bbb R}^d)}.\end{eqnarray*} 利用(2.3)和引理3.2,注意到$d(1/p_{2}-1)<d(1/p_{1}-1)$ 和$\rho(x_{0})>r,$ 则有 \begin{eqnarray*} J_{21}^3&\leq&C\|b\|_{BMO}|B|^{1-1/p_{1}}[\rho(x_{0})]^{d(1/p_{2}-1)-\delta}\int_{0}^{r^{2}}t^{\frac{\alpha+\delta}{2}}\frac{{\rm d}t}{t} \|f(x)\|_{\Lambda_{L}^{1/p_{2}-1}({\Bbb R}^d)}\\ &\leq&C\|b\|_{BMO}\|f(x)\|_{\Lambda_{L}^{1/p_{2}-1}({\Bbb R}^d)}.\end{eqnarray*} 现在估计$J_{21}^2$,记 \begin{eqnarray*} J_{21}^2&\leq&\frac{1}{|B|^{1/p_{1}}}\int_{B}\int_{0}^{r^{2}}\int_{{\Bbb R}^d}t^{\frac{\alpha-d}{2}} e^{-\frac{|x-y|^{2}}{4t}}|(b(y)-b_{B})(f(y)-f_{B(x,r)})\chi_{(4B)^{c}}(y)|{\rm d}y\frac{{\rm d}t}{t}{\rm d}x\\ &&+\frac{1}{|B|^{1/p_{1}}}\int_{B}\int_{0}^{r^{2}}\int_{{\Bbb R}^d}t^{\frac{\alpha-d}{2}} e^{-\frac{|x-y|^{2}}{4t}}|(b(y)-b_{B})(f_{B(x,r)}-f_{B})\chi_{(4B)^{c}}(y)|{\rm d}y\frac{{\rm d}t}{t}{\rm d}x\\ &\triangleq&J_{211}^2(x)+J_{212}^2(x).\end{eqnarray*} 注意到$\rho(x)\sim\rho(x_{0})>r,$ 类似与$J_{21}^3$的估计,有 \[J_{212}^2\leq C\|b\|_{BMO}\|f(x)\|_{\Lambda_{L}^{1/p_{2}-1}(\mathbb{R}^{d})}. \] 由$x\in B,y\in(4B)^{c},$ 知$|x-y|>3r$. 又$\Lambda_{L}^{1/p_{2}-1}({\Bbb R}^d)\subseteq\Lambda^{1/p_{2}-1}({\Bbb R}^d).$ 且设$N'>d/p_{2}$,结合(2.2)式,可证 \begin{eqnarray*} J_{211}^2&\leq& C\frac{1}{|B|^{1/p_{1}}}\int_{B}\int_{0}^{r^{2}}t^{\frac{\alpha-d}{2}} \\ &&\times \sum_{j=1}^\infty\int_{2^{j}r<|x-y|\leq2^{j+1}r}e^{-\frac{|x-y|^{2}}{4t}} |(b(y)-b_{B})(f(y)-f_{B(x,r)})|{\rm d}y\frac{{\rm d}t}{t}{\rm d}x\\ &\leq&C\frac{1}{|B|^{1/p_{1}}}\int_{B}\int_{0}^{r^{2}}t^{\frac{\alpha-d}{2}}\sum_{j=1}^\infty e^{-\frac{A2^{2j}r^{2}}{t}}\int_ {B(x,2^{j+1}r)}|(b(y)-b_{B})(f(y)-f_{B(x,r)})|{\rm d}y\frac{{\rm d}t}{t}{\rm d}x\\ &\leq&C\frac{1}{|B|^{1/p_{1}}}\int_{B}\int_{0}^{r^{2}}t^{\frac{\alpha-d}{2}} \sum_{j=1}^\infty\bigg(\frac{2^{j}r}{\sqrt{t}}\bigg)^{-N'}|B(x,2^{j+1}r)| \\ &&\times \bigg(\frac{1}{|B(x,2^{j+1}r)|}\int_{B(x,2^{j+1}r)}|b(y)-b_{B}|^{s'}{\rm d}y\bigg)^{\frac{1}{s'}}\\ &&\times \bigg(\frac{1}{|B(x,2^{j+1}r)|}\int_{B(x,2^{j+1}r)}|f(y)-f_{B(x,r)}|^{s}{\rm d}y\bigg)^{\frac{1}{s}}\frac{{\rm d}t}{t}{\rm d}x\\ &\leq&C\|b\|_{BMO}r^{-N'+d/p_{2}}\frac{1}{|B|^{1/p_{1}}}\int_{B}\int_{0}^{r^{2}}t^{\frac{\alpha-d+N'}{2}}\frac{{\rm d}t}{t}\sum_{j=1}^\infty j2^{-j(N'-d/p_{2})}\|f(x)\|_{\Lambda^{1/p_{2}-1}({\Bbb R}^d)}{\rm d}x\\ &\leq&C\|b\|_{BMO}\|f\|_{\Lambda_{L}^{1/p_{2}-1}({\Bbb R}^d)}.\end{eqnarray*} 因此 \[J_{211}^2\leq C\|b\|_{BMO}\|f(x)\|_{\Lambda_{L}^{1/p_{2}-1}({\Bbb R}^d)}. \] 且 \[J_{21}^2 \leq C\|b\|_{BMO}\|f(x)\|_{\Lambda_{L}^{1/p_{2}-1}({\Bbb R}^d)}. \] 即 \[J_{2}\leq C\|b\|_{BMO}\|f(x)\|_{\Lambda_{L}^{1/p_{2}-1}({\Bbb R}^d)}.\]

现在估计$J_{1}$. 因为 \begin{eqnarray*} J_{1} &\leq&C\frac{1}{|B|^{1/p_{1}}}\int_{B}|A_{1}f(x)|{\rm d}x\\ &\leq&C\frac{1}{|B|^{1/p_{1}}}\int_{B}|b(y)-b_{B}| \bigg|\int_{0}^{r^{2}}\int_{{\Bbb R}^d} K_{t}^{L}(x,y)f(y){\rm d}yt^{\frac{\alpha}{2}}\frac{{\rm d}t}{t}\bigg|{\rm d}x\\ &\leq&C\frac{1}{|B|^{1/p_{1}}}\int_{B}|b(y)-b_{B}| \bigg|\int_{r^{2}}^{\infty}\int_{{\Bbb R}^d} [K_{t}^{L}(x,y)-K_{t}^{L}(x_{0},y)]f(y){\rm d}yt^{\frac{\alpha}{2}}\frac{{\rm d}t}{t}\bigg|{\rm d}x\\ &\leq&C\frac{1}{|B|^{1/p_{1}}}\int_{B}|b(y)-b_{B}| \bigg|\int_{r^{2}}^{\infty}\int_{{\Bbb R}^d} K_{t}^{L}(x_{0},y)f(y){\rm d}yt^{\frac{\alpha}{2}}\frac{{\rm d}t}{t}\bigg|{\rm d}x\\ &\triangleq&J_{11}+J_{12}+J_{13}.\end{eqnarray*} 对$J_{11}$,由Hölder不等式及定理1.2中$J_{21}$的估计方法有 \[J_{11}\leq C\|b\|_{BMO}\|f(x)\|_{\Lambda_{L}^{1/p_{2}-1}({\Bbb R}^d)}.\] 至于$J_{12}$,类似于定理1.2中$J_{22}$的估计可得 \[J_{12}\leq C\|b\|_{BMO}\|f(x)\|_{\Lambda_{L}^{1/p_{2}-1}({\Bbb R}^d)}.\] 和定理1.2中$J_{12}$的估计得 \[J_{13}\leq C\|b\|_{BMO}\|f(x)\|_{\Lambda_{L}^{1/p_{2}-1}({\Bbb R}^d)}.\] 因此 \[J_{1}\leq C\|b\|_{BMO}\|f(x)\|_{\Lambda_{L}^{1/p_{2}-1}({\Bbb R}^d)}.\] 故(3.6)式得证. 即完成了定理1.2的证明.