分数次极大算子及交换子在加权Morrey空间上的有界性

Boundedness of Fractional Maximal Operator and its Commutator on Weighted Morrey Spaces

摘要: 利用调和分析的实变方法以及权不等式，证明了分数次极大算子在加权Morrey空间上的强有界性和弱有界性，并且得到了分数次极大算子与BMO函数生成的交换子的加权有界性。

Abstract: By applying the real variable methods of harmonic analysis and the weighted inequality, the strong and weak boundedness of the fractional maximal operator is proved on the weighted Morrey space, and the weighted boundedness of commutators generated by fractional maximal operators and BMO functions is obtained.

1. 引言

经典Morrey空间在研究二阶椭圆偏微分方程解的正则性时被首次定义 [1] 。这类函数空间作为经典Lebesgue空间的推广，在调和分析本身及偏微分方程等领域有着重要应用。分数次极大算子作为现代多元调和分析基本的理论工具，其在各类函数空间上的有界性一直受到众多学者的广泛关注。1974年，Muckenhoupt和Wheeden研究了分数次极大算子在加权Lebesgue空间上的有界性 [2] ；1987年，Chiarenza和Frasca获得了分数次极大算子在Morrey空间上的有界性 [3] ；1996年，Ding研究了一类粗糙核极大算子交换子的有界性 [4] ；2016年，Wang和Zhu研究了分数次极大算子在加权变指标空间上的有界性 [5] 。2020年，Duoandikoetxea和Rosenthal引入一类新的加权Morrey空间 [6] ，这类空间是经典Morrey空间的推广并且受到了广泛的关注。2022年，Zhou和Zhao证明了分数次极大算子在这类加权Morrey空间上的有界性 [7] 。受以上启发，本文主要研究分数次极大算子及其交换子在这类加权Morrey空间上的加权估计。关于加权Morrey空间还有更多的结果 [8] [9] [10] [11] [12] 。在叙述本文主要结果之前，首先引入需要的概念和记号。

设 $0\le \alpha <n$ ，f为Rⁿ上局部可积函数，分数次极大算子定义为

$M_{\alpha }f\left(x\right)=\underset{x\in B}{\sup }\frac{1}{{\left|B\right|}^{1-\frac{\alpha }{n}}}{\displaystyle {\int }_B|f\left(y\right)|\text{d}y}$ , (1)

其中，B为Rⁿ中的球体。相应地，给定一个局部可积函数b，由b和 $M_{\alpha }$ 生成的交换子定义为

$\left[b,M_{\alpha }\right]f\left(x\right)=M_{\alpha }\left(bf\right)\left(x\right)-b\left(x\right)M_{\alpha }f\left(x\right)=\underset{x\in B}{\sup }\frac{1}{{|B|}^{1-\frac{\alpha }{n}}}{\displaystyle {\int }_B|f\left(y\right)|\left[b\left(x\right)-b\left(y\right)\right]\text{d}y}$ . (2)

设 $1<p$ ， $q<\infty$ ，Rⁿ上的非负局部可积函数 $w\left(x\right)$ 称为 $A\left(p,q\right)$ 权，如果存在常数 $C>0$ ，使得

$\underset{B}{\sup }{\left(\frac{1}{\left|B\right|}{\displaystyle {\int }_{B}w{\left(x\right)}^q\text{d}x}\right)}^{\frac{1}{q}}{\left(\frac{1}{\left|B\right|}{\displaystyle {\int }_{B}w{\left(x\right)}^{-p^{\prime }}\text{d}x}\right)}^{\frac{1}{p^{\prime }}}\le C<\infty$ . (3)

定义1 [13] BMO空间定义为

$\text{BMO}=\left\{f\in L_{\text{loc}}^1\left(R^n\right):{\Vert f\Vert }_{\text{BMO}}<\infty \right\}$ , (4)

其中

${\Vert f\Vert }_{\text{BMO}}=\underset{x\in B}{\sup }\frac{1}{\left|B\right|}{\displaystyle {\int }_B\left|f\left(x\right)-f_B\right|\text{d}x}$ ,

当 $f\in \text{BMO}$ 时，对任意的 $1\le p<\infty$ ，有

${\Vert f\Vert }_{\text{BMO}}={\Vert f\Vert }_{*}\approx \underset{B}{\sup }{\left(\frac{1}{\left|B\right|}{\displaystyle {\int }_B{\left|f\left(x\right)-f_B\right|}^p\text{d}x}\right)}^{\frac{1}{p}}$ . (5)

定义2 [4] 设 $1\le p<\infty$ ， ${\lambda }_1,{\lambda }_2\in R$ ，f为可测函数，加权Morrey空间定义为

$L^{p,\left({\lambda }_1,{\lambda }_2\right)}\left(w\right)=\left\{f:{\Vert f\Vert }_{L^{p,\left({\lambda }_{1,}{\lambda }_2\right)}\left(w\right)}=\underset{x\in R^n,r>0}{\sup }{\left(\frac{1}{r^{{\lambda }_1}w{\left(B\left(x,r\right)\right)}^{\frac{{\lambda }_2}{n}}}{\displaystyle {\int }_{B\left(x,r\right)}{\left|f\left(y\right)\right|}^{p}w\left(y\right)\text{d}y}\right)}^{\frac{1}{p}}<\infty \right\}$ , (6)

其中， $B\left(x,r\right)$ 表示球心为x、半径为r的球； $w\left(B\left(x,r\right)\right)$ 表示的是非负局部可积函数w在 $B\left(x,r\right)$ 上的积分。

定义3 [4] 设 $1\le p<\infty$ ， ${\lambda }_1,{\lambda }_2\in R$ ，f为可测函数，弱加权Morrey空间定义为

$W{L}^{p,\left({\lambda }_1,{\lambda }_2\right)}\left(w\right)=\left\{f:{\Vert f\Vert }_{W{L}^{p,\left({\lambda }_{1,}{\lambda }_2\right)}\left(w\right)}=\underset{t>0,r>0}{\sup }{\left(\frac{t^{p}w\left(\left\{y\in B\left(x,r\right):|f\left(y\right)|>t\right\}\right)}{r^{{\lambda }_1}w{\left(B\left(x,r\right)\right)}^{\frac{{\lambda }_2}{n}}}\right)}^{\frac{1}{p}}<\infty \right\}$ , (7)

2. 主要定理

定理1 设 $0\le \alpha <n$ ， $M_{\alpha }$ 由式(2)所定义，那么当 $1<p<\frac{\alpha }{n}$ ， $\frac{1}{p}-\frac{1}{q}=\frac{\alpha }{n}$ ，并且 $w\in A\left(p,q\right)$ ， ${\lambda }_2<0$ 时，存在一个与f无关的常数C，使得

${\Vert M_{\alpha }\left(f\right)\Vert }_{L^{q,\left(\frac{q\left({\lambda }_1+{\lambda }_2\right)}{p}-{\lambda }_{2,}{\lambda }_2\right)}\left(w^q\right)}\le C{\Vert f\Vert }_{L^{p,\left({\lambda }_1,{\lambda }_2\right)}\left(w^p\right)}$ .

定理2 设 $0\le \alpha <n$ ， $M_{\alpha }$ 由式(2)所定义，那么当 $1<p<\frac{\alpha }{n}$ ， $\frac{1}{p}-\frac{1}{q}=\frac{\alpha }{n}$ ，并且 $w\in A\left(p,q\right)$ ， ${\lambda }_2<0$ 时，存在一个与f无关的常数C，使得

${\Vert M_{\alpha }\left(f\right)\Vert }_{W{L}^{q,\left(\frac{q\left({\lambda }_1+{\lambda }_2\right)}{p}-{\lambda }_{2,}{\lambda }_2\right)}\left(w^q\right)}\le C{\Vert f\Vert }_{L^{p,\left({\lambda }_1,{\lambda }_2\right)}\left(w^p\right)}$ .

定理3 设 $0\le \alpha <n$ ， ， $\left[b,M_{\alpha }\right]$ 由式(3)所定义，那么当 $1<p_1<\frac{n}{\alpha }$ ， ${p^{\prime }}_1<p_2<\infty$ ， $\frac{1}{q}=\frac{1}{p_1}+\frac{1}{p_2}-\frac{\alpha }{n}$ ， $\frac{1}{t}=\frac{1}{p_1}-\frac{\alpha }{n}$ ， $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$ ，并且 $b\in \text{BMO}\left(R^n\right)$ ， $w\in A\left(p,q\right)$ ， $1+\frac{{\lambda }_2}{qn}\le \frac{1}{p_1}$ 时，存在一个与f无关的常数C，使得

${\Vert \left[b,M_{\alpha }\right]\left(f\right)\Vert }_{L^{q,\left(\frac{q\left({\lambda }_1+{\lambda }_2\right)}{p_1}+\frac{qn}{p_2}-{\lambda }_{2,}{\lambda }_2\right)}\left(w^q\right)}\le C{\Vert b\Vert }_{*}{\Vert f\Vert }_{L^{p_1,\left({\lambda }_1,{\lambda }_2\right)}\left(w^{p_1}\right)}$ .

本文中， $p^{\prime }$ 表示p的共轭，即 $\frac{1}{p}+\frac{1}{p^{\prime }}=1$ ； ${\chi }_E$ 代表集合E的特征函数；C是和主要函数和参量无关的常数，在不同行中甚至在同一行中可以不同。

3. 定理的证明

在证明定理之前，先给出以下引理。

引理1 [8] 设 $0\le \alpha <n$ ， $1<p<\frac{n}{\alpha }$ ， $\frac{1}{p}-\frac{1}{q}=\frac{\alpha }{n}$ ，当 $w\in A\left(p,q\right)$ 时，存在一个与f无关的常数C，使得

${\Vert M_{\alpha }\left(f\right)\Vert }_{L^q\left(w^q\right)}\le C{\Vert f\Vert }_{L^p\left(w^p\right)}$ .

引理2 [8] 设 $0\le \alpha <n$ ， $1\le p<\frac{n}{\alpha }$ ， $\frac{1}{p}-\frac{1}{q}=\frac{\alpha }{n}$ ，当 $w\in A\left(p,q\right)$ 时，对任意的 $\lambda >0$ ，存在一个与f无关的常数C，使得

${\left({\displaystyle {\int }_{\{x\in R^n:M_{\alpha }f\left(x\right)>\lambda \}}w{\left(x\right)}^q\text{d}x}\right)}^{\frac{1}{q}}\le \frac{C}{\lambda }{\left({\displaystyle {\int }_{R^n}{\left|f\left(x\right)w\left(x\right)\right|}^p\text{d}x}\right)}^{\frac{1}{p}}$ .

引理3 [14] 如果 $w\in A_q$ ，则对任意的 $k\in Z^{+}$ ， $l>1$ 以及任意球 $B\subset R^n$ ，当 $1\le q<\infty$ 时，有

$w\left(lB\right)\le C{l}^{nq}w\left(B\right)$ .

定理1的证明 设 $f\in L^{p,\left({\lambda }_1,{\lambda }_2\right)}\left(w^p\right)$ 为局部可积函数，给定任意的球 $B\left(x,r\right)$ ，分解f为 $f=f_1+f_2$ ，其中 $f_1=f{\chi }_{2B}$ ， $f_2=f{\chi }_{{\left(2B\right)}^c}$ ，则

${\Vert M_{\alpha }\left(f\right)\Vert }_{L^{q,\left(\frac{q\left({\lambda }_1+{\lambda }_2\right)}{p}-{\lambda }_{2,}{\lambda }_2\right)}\left(w^q\right)}\le {\Vert M_{\alpha }\left(f_1\right)\Vert }_{L^{q,\left(\frac{q\left({\lambda }_1+{\lambda }_2\right)}{p}-{\lambda }_{2,}{\lambda }_2\right)}\left(w^q\right)}+{\Vert M_{\alpha }\left(f_2\right)\Vert }_{L^{q,\left(\frac{q\left({\lambda }_1+{\lambda }_2\right)}{p}-{\lambda }_{2,}{\lambda }_2\right)}\left(w^q\right)}:=A_1+A_2.$

对于 $A_1$ ，利用引理1，有

$\begin{array}{c}{A}_1={\left(\frac{1}{r^{\frac{q\left({\lambda }_1+{\lambda }_2\right)}{p}-{\lambda }_2}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{n}}}{\displaystyle {\int }_B{\left(M_{\alpha }{f}_1\left(x\right)\right)}^q{w}^q\left(x\right)\text{d}x}\right)}^{\frac{1}{q}}\\ \le C{\left(\frac{1}{r^{\frac{q\left({\lambda }_1+{\lambda }_2\right)}{p}-{\lambda }_2}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{n}}}{\displaystyle {\int }_{R^n}{\left(M_{\alpha }{f}_1\left(x\right)\right)}^q{w}^q\left(x\right)\text{d}x}\right)}^{\frac{1}{q}}\\ \le C{\left(\frac{1}{r^{\frac{q\left({\lambda }_1+{\lambda }_2\right)}{p}-{\lambda }_2}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{n}}}\right)}^{\frac{1}{q}}{\left({\displaystyle {\int }_{R^n}{f}_1{\left(x\right)}^p{w}^p\left(x\right)\text{d}x}\right)}^{\frac{1}{p}}\\ \simeq C{\left(\frac{1}{r^{\frac{q\left({\lambda }_1+{\lambda }_2\right)}{p}-{\lambda }_2}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{n}}}\right)}^{\frac{1}{q}}{\left({\displaystyle {\int }_{2B}f{\left(x\right)}^p{w}^p\left(x\right)\text{d}x}\right)}^{\frac{1}{p}}\\ \le C\frac{r^{\frac{{\lambda }_1}{p}}{w}^p{\left(2B\right)}^{\frac{{\lambda }_2}{np}}}{{\left(r^{\frac{q\left({\lambda }_1+{\lambda }_2\right)}{p}-{\lambda }_2}{w}^q{\left(B\right)}^{\frac{{\lambda }_2}{n}}\right)}^{\frac{1}{q}}}{\Vert f\Vert }_{L^{p,\left({\lambda }_1,{\lambda }_2\right)}\left(w^p\right)}.\end{array}$

利用Hölder不等式，当 $1\le p<q<\infty$ 时，有

$w^p\left(B\right)\le {\left(w^q\left(B\right)\right)}^{\frac{p}{q}}{\left|B\right|}^{1-\frac{p}{q}}$ . (8)

通过式(8)，不难得到

$w^p\left(2B\right)\le {\left(w^q\left(2B\right)\right)}^{\frac{p}{q}}{r}^{n\left(1-\frac{p}{q}\right)}$ . (9)

显然

$\frac{r^{\frac{{\lambda }_1}{p}}{w}^p{\left(2B\right)}^{\frac{{\lambda }_2}{np}}}{{\left(r^{\frac{q\left({\lambda }_1+{\lambda }_2\right)}{p}-{\lambda }_2}{w}^q{\left(B\right)}^{\frac{{\lambda }_2}{n}}\right)}^{\frac{1}{q}}}\le {\left(\frac{w^q\left(2B\right)}{w^q\left(B\right)}\right)}^{\frac{{\lambda }_2}{qn}}\le 1$ , (10)

所以

$A_1\le C{\Vert f\Vert }_{L^{p,\left({\lambda }_1,{\lambda }_2\right)}\left(w^p\right)}$ . (11)

同理，对于 $A_2$ ，利用引理1，有

$\begin{array}{c}{A}_2={\left(\frac{1}{r^{\frac{q\left({\lambda }_1+{\lambda }_2\right)}{p}-{\lambda }_2}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{n}}}{\displaystyle {\int }_B{\left(M_{\alpha }{f}_2\left(x\right)\right)}^q{w}^q\left(x\right)\text{d}x}\right)}^{\frac{1}{q}}\\ \le C{\left(\frac{1}{r^{\frac{q\left({\lambda }_1+{\lambda }_2\right)}{p}-{\lambda }_2}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{n}}}{\displaystyle {\int }_{R^n}{\left(M_{\alpha }{f}_2\left(x\right)\right)}^q{w}^q\left(x\right)\text{d}x}\right)}^{\frac{1}{q}}\\ \le C{\left(\frac{1}{r^{\frac{q\left({\lambda }_1+{\lambda }_2\right)}{p}-{\lambda }_2}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{n}}}\right)}^{\frac{1}{q}}{\left({\displaystyle {\int }_{R^n}{f}_2{\left(x\right)}^p{w}^p\left(x\right)\text{d}x}\right)}^{\frac{1}{p}}\\ \simeq C{\left(\frac{1}{r^{\frac{q\left({\lambda }_1+{\lambda }_2\right)}{p}-{\lambda }_2}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{n}}}\right)}^{\frac{1}{q}}{\left({\displaystyle {\int }_{{\left(2B\right)}^c}f{\left(x\right)}^p{w}^p\left(x\right)\text{d}x}\right)}^{\frac{1}{p}}\end{array}$

$\begin{array}{l}\le C{\left(\frac{1}{r^{\frac{q\left({\lambda }_1+{\lambda }_2\right)}{p}-{\lambda }_2}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{n}}}\right)}^{\frac{1}{q}}{\left(\underset{j=1}{\stackrel{\infty }{{{\displaystyle \sum }}^{\text{}}}}{\displaystyle {\int }_{2^{j+1}B\backslash 2^{j}B}f{\left(x\right)}^p{w}^p\left(x\right)\text{d}x}\right)}^{\frac{1}{p}}\\ \le C{\left(\frac{1}{r^{\frac{q\left({\lambda }_1+{\lambda }_2\right)}{p}-{\lambda }_2}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{n}}}\right)}^{\frac{1}{q}}{\left(\underset{j=1}{\stackrel{\infty }{{{\displaystyle \sum }}^{\text{}}}}{\displaystyle {\int }_{2^{j+1}B}f{\left(x\right)}^p{w}^p\left(x\right)\text{d}x}\right)}^{\frac{1}{p}}\\ \le C\frac{r^{\frac{{\lambda }_1}{p}}\underset{j=1}{\stackrel{\infty }{{{\displaystyle \sum }}^{\text{}}}}{w}^p{\left(2^{j+1}B\right)}^{\frac{{\lambda }_2}{np}}}{{\left(r^{\frac{q\left({\lambda }_1+{\lambda }_2\right)}{p}-{\lambda }_2}{w}^q{\left(B\right)}^{\frac{{\lambda }_2}{n}}\right)}^{\frac{1}{q}}}{\Vert f\Vert }_{L^{p,\left({\lambda }_1,{\lambda }_2\right)}\left(w^p\right)}\\ \le {\left(\frac{\underset{j=1}{\stackrel{\infty }{{{\displaystyle \sum }}^{\text{}}}}{w}^q\left(2^{j+1}B\right)}{w^q\left(B\right)}\right)}^{\frac{{\lambda }_2}{qn}}{\Vert f\Vert }_{L^{p,\left({\lambda }_1,{\lambda }_2\right)}\left(w^p\right)}.\end{array}$

利用 ${\lambda }_2<0$ 、引理3，有

$A_2\le C{\Vert f\Vert }_{L^{p,\left({\lambda }_1,{\lambda }_2\right)}\left(w^p\right)}$ . (12)

结合式(11)、(12)，定理1证毕。

定理2的证明 结合定理1的证明，我们只需考虑 $p=1$ 的情况便可证明定理2。对局部可积函数 $f\in L^{1,\left({\lambda }_1,{\lambda }_2\right)}\left(w\right)$ ，给定任意的球 $B\left(x_0,r\right)$ ，分解f为 $f=f_1+f_2$ ，其中 $f_1=f{\chi }_{2B}$ ， $f_2=f{\chi }_{{\left(2B\right)}^c}$ ，则 $M_{\alpha }f\left(x\right)\le M_{\alpha }{f}_1\left(x\right)+M_{\alpha }{f}_2\left(x\right)$ ，从而

${\displaystyle {\int }_{\{x\in B:M_{\alpha }f\left(x\right)>t\}}{w}^q\left(x\right)\text{d}x}\le {\displaystyle {\int }_{\left\{x\in B:M_{\alpha }{f}_1\left(x\right)>\frac{t}{2}\right\}}{w}^q\left(x\right)\text{d}x}+{\displaystyle {\int }_{\left\{x\in B:M_{\alpha }{f}_2\left(x\right)>\frac{t}{2}\right\}}{w}^q\left(x\right)\text{d}x}$ ,

显然，

$\begin{array}{c}{\Vert M_{\alpha }\left(f\right)\Vert }_{W{L}^{q,\left(q\left({\lambda }_1+{\lambda }_2\right)-{\lambda }_2{}_{,}{\lambda }_2\right)}\left(w^q\right)}={\left(\frac{t^q{{\displaystyle \int }}_{\{x\in B:M_{\alpha }f\left(x\right)>t\}}{w}^q\left(x\right)\text{d}x}{r^{q\left({\lambda }_1+{\lambda }_2\right)-{\lambda }_2}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{n}}}\right)}^{\frac{1}{q}}\\ \le C{\left(\frac{t^q{{\displaystyle \int }}_{\left\{x\in B:M_{\alpha }{f}_1\left(x\right)>\frac{t}{2}\right\}}{w}^q\left(x\right)\text{d}x}{r^{q\left({\lambda }_1+{\lambda }_2\right)-{\lambda }_2}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{n}}}\right)}^{\frac{1}{q}}+C{\left(\frac{t^q{{\displaystyle \int }}_{\left\{x\in B:M_{\alpha }{f}_2\left(x\right)>\frac{t}{2}\right\}}{w}^q\left(x\right)\text{d}x}{r^{q\left({\lambda }_1+{\lambda }_2\right)-{\lambda }_2}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{n}}}\right)}^{\frac{1}{q}}\\ :=B_1+B_2.\end{array}$

对于 $B_1$ ，利用引理2，有

$\begin{array}{c}{B}_1\le C{\left(\frac{t^q{{\displaystyle \int }}_{\left\{x\in R^n:M_{\alpha }{f}_1\left(x\right)>\frac{t}{2}\right\}}{w}^q\left(x\right)\text{d}x}{r^{q\left({\lambda }_1+{\lambda }_2\right)-{\lambda }_2}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{n}}}\right)}^{\frac{1}{q}}\\ \le C{\left(\frac{{\left({{\displaystyle \int }}_{R^n}{f}_1\left(x\right)w\left(x\right)\text{d}x\right)}^q}{r^{q\left({\lambda }_1+{\lambda }_2\right)-{\lambda }_2}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{n}}}\right)}^{\frac{1}{q}}\\ \simeq \frac{{{\displaystyle \int }}_{2B}f\left(x\right)w\left(x\right)\text{d}x}{r^{\left({\lambda }_1+{\lambda }_2\right)-\frac{{\lambda }_2}{q}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}\end{array}$

$\begin{array}{l}\le C\frac{r^{{\lambda }_1}{\left({{\displaystyle \int }}_{2B}w\left(x\right)\text{d}x\right)}^{\frac{{\lambda }_2}{n}}}{r^{\left({\lambda }_1+{\lambda }_2\right)-\frac{{\lambda }_2}{q}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}{\Vert f\Vert }_{L^{1,\left({\lambda }_1,{\lambda }_2\right)}\left(w\right)}\\ \le C\frac{r^{{\lambda }_1}{\left(w\left(2B\right)\right)}^{\frac{{\lambda }_2}{n}}}{r^{\left({\lambda }_1+{\lambda }_2\right)-\frac{{\lambda }_2}{q}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}{\Vert f\Vert }_{L^{1,\left({\lambda }_1,{\lambda }_2\right)}\left(w\right)}\\ \le C\frac{r^{\left({\lambda }_1+{\lambda }_2\right)-\frac{{\lambda }_2}{q}}{\left(w^q\left(2B\right)\right)}^{\frac{{\lambda }_2}{qn}}}{r^{\left({\lambda }_1+{\lambda }_2\right)-\frac{{\lambda }_2}{q}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}{\Vert f\Vert }_{L^{1,\left({\lambda }_1,{\lambda }_2\right)}\left(w\right)}。\end{array}$

利用 ${\lambda }_2<0$ 、引理3，得到

$B_1\le C{\Vert f\Vert }_{L^{p_{1,\left({\lambda }_1,{\lambda }_2\right)}}\left(w\right)}$ . (13)

对于 $B_2$ ，利用引理2，有

$\begin{array}{c}{B}_2\le C{\left(\frac{t^q{{\displaystyle \int }}_{\left\{x\in R^n:M_{\alpha }{f}_2\left(x\right)>\frac{t}{2}\right\}}{w}^q\left(x\right)\text{d}x}{r^{q\left({\lambda }_1+{\lambda }_2\right)-{\lambda }_2}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{n}}}\right)}^{\frac{1}{q}}\\ \le C{\left(\frac{{\left({{\displaystyle \int }}_{R^n}{f}_2\left(x\right)w\left(x\right)\text{d}x\right)}^q}{r^{q\left({\lambda }_1+{\lambda }_2\right)-{\lambda }_2}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{n}}}\right)}^{\frac{1}{q}}\\ \simeq \frac{{{\displaystyle \int }}_{{\left(2B\right)}^c}f\left(x\right)w\left(x\right)\text{d}x}{r^{\left({\lambda }_1+{\lambda }_2\right)-\frac{{\lambda }_2}{q}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}\\ \le C\frac{r^{{\lambda }_1}{\left(\underset{j=1}{\stackrel{\infty }{{{\displaystyle \sum }}^{\text{}}}}{{\displaystyle \int }}_{2^{j+1}B\backslash 2^{j}B}w\left(x\right)\text{d}x\right)}^{\frac{{\lambda }_2}{n}}}{r^{\left({\lambda }_1+{\lambda }_2\right)-\frac{{\lambda }_2}{q}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}{\Vert f\Vert }_{L^{1,\left({\lambda }_1,{\lambda }_2\right)}(\; w\; )}\end{array}$

$\begin{array}{l}\le C\frac{r^{{\lambda }_1}{\left(\underset{j=1}{\stackrel{\infty }{{{\displaystyle \sum }}^{\text{}}}}{{\displaystyle \int }}_{2^{j+1}B}w\left(x\right)\text{d}x\right)}^{\frac{{\lambda }_2}{n}}}{r^{\left({\lambda }_1+{\lambda }_2\right)-\frac{{\lambda }_2}{q}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}{\Vert f\Vert }_{L^{1,\left({\lambda }_1,{\lambda }_2\right)}\left(w\right)}\\ \le C\frac{r^{{\lambda }_1}{\left(\underset{j=1}{\stackrel{\infty }{{{\displaystyle \sum }}^{\text{}}}}w\left(2^{j+1}B\right)\right)}^{\frac{{\lambda }_2}{n}}}{r^{\left({\lambda }_1+{\lambda }_2\right)-\frac{{\lambda }_2}{q}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}{\Vert f\Vert }_{L^{1,\left({\lambda }_1,{\lambda }_2\right)}\left(w\right)}\\ \le C\frac{r^{\left({\lambda }_1+{\lambda }_2\right)-\frac{{\lambda }_2}{q}}{\left(\underset{j=1}{\stackrel{\infty }{{{\displaystyle \sum }}^{\text{}}}}{w}^q\left(2^{j+1}B\right)\right)}^{\frac{{\lambda }_2}{qn}}}{r^{\left({\lambda }_1+{\lambda }_2\right)-\frac{{\lambda }_2}{q}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}{\Vert f\Vert }_{L^{1,\left({\lambda }_1,{\lambda }_2\right)}\left(w\right)}.\end{array}$

利用 ${\lambda }_2<0$ 、引理3，得到

$B_2\le C{\Vert f\Vert }_{L^{p_{1,\left({\lambda }_1,{\lambda }_2\right)}}\left(w\right)}$ . (14)

结合式(13)、(14)，定理2证毕。

定理3的证明 同理，分解f为 $f=f_1+f_2$ ，其中 $f_1=f{\chi }_{2B}$ ， $f_2=f{\chi }_{{\left(2B\right)}^c}$ ，则

$\begin{array}{c}{\Vert \left[b,M_{\alpha }\right]\left(f\right)\left(x\right)\Vert }_{L^{q,\left(\frac{q\left({\lambda }_1+{\lambda }_2\right)}{p_1}+\frac{qn}{p_2}-{\lambda }_2,{\lambda }_2\right)}\left(w^q\right)}={\left(\frac{1}{r^{\frac{q\left({\lambda }_1+{\lambda }_2\right)}{p_1}-{\lambda }_2+\frac{qn}{p_2}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{n}}}{\displaystyle {\int }_B{\left|\left[b,M_{\alpha }\left(f\right)\left(x\right)\right]\right|}^q{w}^q\left(x\right)\text{d}x}\right)}^{\frac{1}{q}}\\ \le \frac{1}{r^{\frac{\left({\lambda }_1+{\lambda }_2\right)}{p_1}-\frac{{\lambda }_2}{q}+\frac{n}{p_2}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}{\left({\displaystyle {\int }_B{\left|\left[b,M_{\alpha }\left(f_1\right)\left(x\right)\right]\right|}^q{w}^q\left(x\right)\text{d}x}\right)}^{\frac{1}{q}}\\ +\frac{1}{r^{\frac{\left({\lambda }_1+{\lambda }_2\right)}{p_1}-\frac{{\lambda }_2}{q}+\frac{n}{p_2}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}{\left({\displaystyle {\int }_B{\left|\left[b,M_{\alpha }\left(f_2\right)\left(x\right)\right]\right|}^q{w}^q\left(x\right)\text{d}x}\right)}^{\frac{1}{q}}\\ :=E_1+E_2.\end{array}$

对于 $E_1$ ，有

$\begin{array}{l}{E}_1\le \frac{1}{r^{\frac{\left({\lambda }_1+{\lambda }_2\right)}{p_1}-\frac{{\lambda }_2}{q}+\frac{n}{p_2}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}{\left({\displaystyle {\int }_B{\left|b\left(x\right)-b_B\right|}^q{\left|M_{\alpha }\left(f_1\right)\left(x\right)\right|}^q{w}^q\left(x\right)\text{d}x}\right)}^{\frac{1}{q}}\\ +\frac{1}{r^{\frac{\left({\lambda }_1+{\lambda }_2\right)}{p_1}-\frac{{\lambda }_2}{q}+\frac{n}{p_2}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}{\left({\displaystyle {\int }_B{\left|M_{\alpha }\left(f_1\left(b-b_B\right)\right)\left(x\right)\right|}^q{w}^q\left(x\right)\text{d}x}\right)}^{\frac{1}{q}}:=E_{11}+E_{12}.\end{array}$

对于 $E_{11}$ ，利用Hölder不等式、引理1，有

$\begin{array}{c}{E}_{11}\le \frac{1}{r^{\frac{\left({\lambda }_1+{\lambda }_2\right)}{p_1}-\frac{{\lambda }_2}{q}+\frac{n}{p_2}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}{\left({\displaystyle {\int }_B{\left|b\left(x\right)-b_B\right|}^{p_2}\text{d}x}\right)}^{\frac{1}{p_2}}{\left({\displaystyle {\int }_B{\left|M_{\alpha }\left(f_1\right)\left(x\right)\right|}^t{w}^t\left(x\right)\text{d}x}\right)}^{\frac{1}{t}}\\ \le \frac{{\left|B\right|}^{\frac{1}{p_2}}}{r^{\frac{\left({\lambda }_1+{\lambda }_2\right)}{p_1}-\frac{{\lambda }_2}{q}+\frac{n}{p_2}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}{\left(\frac{1}{\left|B\right|}{\displaystyle {\int }_B{\left|b\left(x\right)-b_B\right|}^{p_2}\text{d}x}\right)}^{\frac{1}{p_2}}{\left({\displaystyle {\int }_B{\left|M_{\alpha }\left(f_1\right)\left(x\right)\right|}^t{w}^t\left(x\right)\text{d}x}\right)}^{\frac{1}{t}}\\ \le C\frac{1}{r^{\frac{\left({\lambda }_1+{\lambda }_2\right)}{p_1}-\frac{{\lambda }_2}{q}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}{\Vert b\Vert }_{*}{\left({\displaystyle {\int }_{2B}{\left|f\left(x\right)\right|}^{p_1}{w}^{p_1}\left(x\right)\text{d}x}\right)}^{\frac{1}{p_1}}\\ \le C{\Vert b\Vert }_{*}{\Vert f\Vert }_{L^{p_{1,\left({\lambda }_1,{\lambda }_2\right)}}\left(w^{p_1}\right)}\frac{r^{\frac{{\lambda }_1}{p_1}}{w}^{p_1}{\left(2B\right)}^{\frac{{\lambda }_2}{n{p}_1}}}{r^{\frac{\left({\lambda }_1+{\lambda }_2\right)}{p_1}-\frac{{\lambda }_2}{q}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}\\ \le C{\Vert b\Vert }_{*}{\Vert f\Vert }_{L^{p_{1,\left({\lambda }_1,{\lambda }_2\right)}}\left(w^{p_1}\right)}\frac{w^q{\left(2B\right)}^{\frac{{\lambda }_2}{qn}}}{w^q{\left(B\right)}^{\frac{{\lambda }_2}{qn}}}.\end{array}$

利用 ${\lambda }_2<0$ 、引理3，得到

$E_{11}\le C{\Vert b\Vert }_{*}{\Vert f\Vert }_{L^{p_{1,\left({\lambda }_1,{\lambda }_2\right)}}\left(w^{p_1}\right)}$ . (15)

对于 $E_{12}$ ，利用引理1、Hölder不等式，有

$\begin{array}{c}{E}_{12}\le \frac{1}{r^{\frac{\left({\lambda }_1+{\lambda }_2\right)}{p_1}-\frac{{\lambda }_2}{q}+\frac{n}{p_2}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}{\left({\displaystyle {\int }_{2B}{\left|\left(b\left(x\right)-b_B\right)f\left(x\right)\right|}^p{w}^p\left(x\right)\text{d}x}\right)}^{\frac{1}{p}}\\ \le C\frac{1}{r^{\frac{\left({\lambda }_1+{\lambda }_2\right)}{p_1}-\frac{{\lambda }_2}{q}+\frac{n}{p_2}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}{\left({\displaystyle {\int }_{2B}{\left|b\left(x\right)-b_B\right|}^{p_2}\text{d}x}\right)}^{\frac{1}{p_2}}{\left({\displaystyle {\int }_{2B}{\left|f\left(x\right)\right|}^{p_1}{w}^{p_1}\left(x\right)\text{d}x}\right)}^{\frac{1}{p_1}}\\ \le C\frac{r^{\frac{{\lambda }_1}{p_1}}{w}^{p_1}{\left(2B\right)}^{\frac{{\lambda }_2}{n{p}_1}}}{r^{\frac{\left({\lambda }_1+{\lambda }_2\right)}{p_1}-\frac{{\lambda }_2}{q}+\frac{n}{p_2}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}{\left({\displaystyle {\int }_{2B}{\left|b\left(x\right)-b_B\right|}^{p_2}\text{d}x}\right)}^{\frac{1}{p_2}}{\Vert f\Vert }_{L^{p_{1,\left({\lambda }_1,{\lambda }_2\right)}}\left(w^{p_1}\right)}\end{array}$

$\begin{array}{l}\le C\frac{r^{\frac{{\lambda }_1}{p_1}}{w}^{p_1}{\left(2B\right)}^{\frac{{\lambda }_2}{n{p}_1}}}{r^{\frac{\left({\lambda }_1+{\lambda }_2\right)}{p_1}-\frac{{\lambda }_2}{q}+\frac{n}{p_2}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}{\Vert f\Vert }_{L^{p_{1,\left({\lambda }_1,{\lambda }_2\right)}}\left(w^{p_1}\right)}\left({\left({\displaystyle {\int }_{2B}{\left|b\left(x\right)-b_{2B}\right|}^{p_2}\text{d}x}\right)}^{\frac{1}{p_2}}+\left|b_{2B}-b_B\right|{\left|2B\right|}^{\frac{1}{p_2}}\right)\\ \le C\frac{r^{\frac{{\lambda }_1}{p_1}}{w}^{p_1}{\left(2B\right)}^{\frac{{\lambda }_2}{n{p}_1}}{\left|2B\right|}^{\frac{1}{p_2}}}{r^{\frac{\left({\lambda }_1+{\lambda }_2\right)}{p_1}-\frac{{\lambda }_2}{q}+\frac{n}{p_2}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}{\Vert f\Vert }_{L^{p_{1,\left({\lambda }_1,{\lambda }_2\right)}}\left(w^{p_1}\right)}\left({\left(\frac{1}{\left|2B\right|}{\displaystyle {\int }_{2B}{\left|b\left(x\right)-b_{2B}\right|}^{p_2}\text{d}x}\right)}^{\frac{1}{p_2}}+\left|b_{2B}-b_B\right|\right)\\ \le C\frac{w^q{\left(2B\right)}^{\frac{{\lambda }_2}{qn}}}{w^q{\left(B\right)}^{\frac{{\lambda }_2}{qn}}}{\Vert f\Vert }_{L^{p_{1,\left({\lambda }_1,{\lambda }_2\right)}}\left(w^{p_1}\right)}{\Vert b\Vert }_{*}.\end{array}$

注意到 ${\lambda }_2<0$ ，结合引理3，有

$E_{12}\le C{\Vert b\Vert }_{*}{\Vert f\Vert }_{L^{p_{1,\left({\lambda }_1,{\lambda }_2\right)}}\left(w^{p_1}\right)}$ . (16)

对于 $E_2$ ，首先

$\begin{array}{c}{\left|\left[b,M_{\alpha }\right]\left(f_2\right)\left(x\right)\right|}^q\le C{\left(\frac{1}{{\left|B\right|}^{1-\frac{\alpha }{n}}}{\displaystyle {\int }_B\left|b\left(x\right)-b\left(y\right)\right|\left|f_2\left(y\right)\right|\text{d}y}\right)}^q\\ \le C{\left(\frac{1}{{\left|B\right|}^{1-\frac{\alpha }{n}}}{\displaystyle {\int }_B\left|f_2\left(y\right)\right|\left(\left|b\left(x\right)-b_B\right|+\left|b_B-b\left(y\right)\right|\right)\text{d}y}\right)}^q\\ \le C{\left(\frac{1}{{\left|B\right|}^{1-\frac{\alpha }{n}}}{\displaystyle {\int }_B\left|f_2\left(y\right)\right|\text{d}y}\right)}^q{\left|b\left(x\right)-b_B\right|}^q+C{\left(\frac{1}{{\left|B\right|}^{1-\frac{\alpha }{n}}}{\displaystyle {\int }_B\left|f_2\left(y\right)\right|\left|b\left(y\right)-b_B\right|\text{d}y}\right)}^q，\end{array}$

因此，有

$\begin{array}{l}{E}_2\le C\frac{1}{r^{\frac{\left({\lambda }_1+{\lambda }_2\right)}{p_1}-\frac{{\lambda }_2}{q}+\frac{n}{p_2}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}{\left({\displaystyle {\int }_B{\left(\frac{1}{{\left|B\right|}^{1-\frac{\alpha }{n}}}\underset{B}{{\displaystyle \int }}\left|f_2\left(y\right)\right|\text{d}y\right)}^q{\left|b\left(x\right)-b_B\right|}^q{w}^q\left(x\right)\text{d}x}\right)}^{\frac{1}{q}}\\ +C\frac{1}{r^{\frac{\left({\lambda }_1+{\lambda }_2\right)}{p_1}-\frac{{\lambda }_2}{q}+\frac{n}{p_2}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}{\left({\displaystyle {\int }_B{\left(\frac{1}{{\left|B\right|}^{1-\frac{\alpha }{n}}}\underset{B}{{\displaystyle \int }}\left|f_2\left(y\right)\right|\left|b\left(y\right)-b_B\right|\text{d}y\right)}^q{w}^q\left(x\right)\text{d}x}\right)}^{\frac{1}{q}}:=E_{21}+E_{22}。\end{array}$

对于 $E_{21}$ ，利用Minkowski不等式、Hölder不等式，有

$\begin{array}{c}{E}_{21}\le C\frac{1}{r^{\frac{\left({\lambda }_1+{\lambda }_2\right)}{p_1}-\frac{{\lambda }_2}{q}+\frac{n}{p_2}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}{\left({\displaystyle {\int }_B{\left(\frac{1}{{\left|B\right|}^{1-\frac{\alpha }{n}}}\underset{B}{{\displaystyle \int }}\left|f_2\left(y\right)\right|\text{d}y\right)}^q{\left|b\left(x\right)-b_B\right|}^q{w}^q\left(x\right)\text{d}x}\right)}^{\frac{1}{q}}\\ \le C\frac{\frac{1}{{\left|B\right|}^{1-\frac{\alpha }{n}}}}{r^{\frac{\left({\lambda }_1+{\lambda }_2\right)}{p_1}-\frac{{\lambda }_2}{q}+\frac{n}{p_2}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}\left[\underset{x\in B}{\sup }w\left(x\right)\right]\underset{j=1}{\stackrel{\infty }{{{\displaystyle \sum }}^{\text{}}}}{\displaystyle {\int }_{2^{j+1}B\backslash 2^{j}B}\left|f\left(y\right)\right|{\left(\underset{B}{{\displaystyle \int }}{\left|b\left(x\right)-b_B\right|}^q\text{d}x\right)}^{\frac{1}{q}}\text{d}y}\\ \le C\frac{\frac{1}{{\left|B\right|}^{1-\frac{\alpha }{n}}}}{r^{\frac{\left({\lambda }_1+{\lambda }_2\right)}{p_1}-\frac{{\lambda }_2}{q}+\frac{n}{p_2}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}\left[\underset{x\in B}{\inf }w\left(x\right)\right]\underset{j=1}{\stackrel{\infty }{{{\displaystyle \sum }}^{\text{}}}}{\displaystyle {\int }_{2^{j+1}B}\left|f\left(y\right)\right|{\left(\underset{B}{{\displaystyle \int }}{\left|b\left(x\right)-b_B\right|}^{p_2}\text{d}x\right)}^{\frac{1}{p_2}}{\left(\underset{B}{{\displaystyle \int }}\text{d}x\right)}^{\frac{1}{t}}\text{d}y}\end{array}$

$\begin{array}{c}\le C\frac{{\left|B\right|}^{\frac{1}{p_2}}}{r^{\frac{\left({\lambda }_1+{\lambda }_2\right)}{p_1}-\frac{{\lambda }_2}{q}+\frac{n}{p_2}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}\frac{1}{{\left|B\right|}^{1-\frac{\alpha }{n}}}\frac{w\left(B\right)}{\left|B\right|}{\Vert b\Vert }_{*}\underset{j=1}{\stackrel{\infty }{{{\displaystyle \sum }}^{\text{}}}}{\displaystyle {\int }_{2^{j+1}B}{\left|B\right|}^{\frac{1}{t}}\left|f\left(y\right)\right|\text{d}y}\\ \le C\frac{1}{r^{\frac{\left({\lambda }_1+{\lambda }_2\right)}{p_1}-\frac{{\lambda }_2}{q}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}\frac{{\left|B\right|}^{\frac{1}{q}-\frac{1}{t}-\frac{1}{p_2}}}{{\left|B\right|}^{1-\frac{\alpha }{n}}}{\Vert b\Vert }_{*}\underset{j=1}{\stackrel{\infty }{{{\displaystyle \sum }}^{\text{}}}}\frac{w\left(2^{j+1}B\right)}{\left|2^{j+1}B\right|}{\left|2^{j+1}B\right|}^{\frac{1}{t}}{\displaystyle {\int }_{2^{j+1}B}\left|f\left(y\right)\right|\text{d}y}\\ \le C\frac{1}{r^{\frac{\left({\lambda }_1+{\lambda }_2\right)}{p_1}-\frac{{\lambda }_2}{q}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}\frac{{\left|B\right|}^{\frac{1}{q}-\frac{1}{t}-\frac{1}{p_2}}}{{\left|B\right|}^{1-\frac{\alpha }{n}}}\underset{j=1}{\stackrel{\infty }{{{\displaystyle \sum }}^{\text{}}}}{\Vert b\Vert }_{*}{\left|B\right|}^{\frac{1}{t}}{\left({\displaystyle {\int }_{2^{j+1}B}{\left|f\left(y\right)\right|}^{p_1}{w}^{p_1}\left(y\right)\text{d}y}\right)}^{\frac{1}{p_1}}{\left|2^{j+1}B\right|}^{1-\frac{1}{p_1}}\end{array}$

$\begin{array}{l}\le C\underset{j=1}{\stackrel{\infty }{{{\displaystyle \sum }}^{\text{}}}}\frac{r^{\frac{{\lambda }_1}{p_1}}{w}^{p_1}{\left(2^{j+1}B\right)}^{\frac{{\lambda }_2}{n{p}_1}}}{r^{\frac{\left({\lambda }_1+{\lambda }_2\right)}{p_1}-\frac{{\lambda }_2}{q}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}{\Vert b\Vert }_{*}\frac{{\left|B\right|}^{\frac{1}{q}-\frac{1}{p_2}}}{{\left|B\right|}^{1-\frac{\alpha }{n}}}{\left|2^{j+1}B\right|}^{1-\frac{1}{p_1}}{\left(\frac{1}{r^{{\lambda }_1}{w}^{p_1}{\left(2^{j+1}B\right)}^{\frac{{\lambda }_2}{n}}}{\displaystyle {\int }_{2^{j+1}B}{\left|f\left(y\right)\right|}^{p_1}{w}^{p_1}\left(y\right)\text{d}y}\right)}^{\frac{1}{p_1}}\\ \le C{\Vert b\Vert }_{*}{\Vert f\Vert }_{L^{p_{1,\left({\lambda }_1,{\lambda }_2\right)}}\left(w^{p_1}\right)}\underset{j=1}{\stackrel{\infty }{{{\displaystyle \sum }}^{\text{}}}}\frac{w^q{\left(2^{j+1}B\right)}^{\frac{{\lambda }_2}{qn}}}{w^q{\left(B\right)}^{\frac{{\lambda }_2}{qn}}}\frac{{\left|B\right|}^{\frac{1}{q}-\frac{1}{p_2}}}{{\left|B\right|}^{1-\frac{\alpha }{n}}}{\left|2^{j+1}B\right|}^{1-\frac{1}{p_1}}\\ \le C{\Vert b\Vert }_{*}{\Vert f\Vert }_{L^{p_{1,\left({\lambda }_1,{\lambda }_2\right)}}\left(w^{p_1}\right)}\underset{j=1}{\stackrel{\infty }{{{\displaystyle \sum }}^{\text{}}}}{\left|2^{j+1}\right|}^{1-\frac{1}{p_1}+\frac{{\lambda }_2}{qn}}{\left|B\right|}^{\frac{1}{q}-\frac{1}{p_1}-\frac{1}{p_2}+\frac{\alpha }{n}}\underset{\begin{array}{l}\\ \end{array}}{\overset{\begin{array}{l}\\ \end{array}}{}}\\ \le C{\Vert b\Vert }_{*}{\Vert f\Vert }_{L^{p_{1,\left({\lambda }_1,{\lambda }_2\right)}}\left(w^{p_1}\right)}\underset{j=1}{\stackrel{\infty }{{{\displaystyle \sum }}^{\text{}}}}{\left|2^{j+1}\right|}^{1-\frac{1}{p_1}+\frac{{\lambda }_2}{qn}}。\end{array}$

利用 $1+\frac{{\lambda }_2}{qn}\le \frac{1}{p_1}$ ，得

$E_{21}\le C{\Vert b\Vert }_{*}{\Vert f\Vert }_{L^{p_{1,\left({\lambda }_1,{\lambda }_2\right)}}}$ . (17)

同理，对于 $E_{22}$ ，利用Minkowski不等式、Hölder不等式，有

$\begin{array}{c}{E}_{22}\le C\frac{1}{r^{\frac{\left({\lambda }_1+{\lambda }_2\right)}{p_1}-\frac{{\lambda }_2}{q}+\frac{n}{p_2}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}{\left({\displaystyle {\int }_B{\left(\frac{1}{{\left|B\right|}^{1-\frac{\alpha }{n}}}\underset{B}{{\displaystyle \int }}\left|f_2\left(y\right)\right|\left|b\left(y\right)-b_B\right|\text{d}y\right)}^q{w}^q\left(x\right)\text{d}x}\right)}^{\frac{1}{q}}\\ \le C\frac{1}{r^{\frac{\left({\lambda }_1+{\lambda }_2\right)}{p_1}-\frac{{\lambda }_2}{q}+\frac{n}{p_2}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}\frac{1}{{\left|B\right|}^{1-\frac{\alpha }{n}}}\underset{j=1}{\stackrel{\infty }{{{\displaystyle \sum }}^{\text{}}}}{\displaystyle {\int }_{2^{j+1}B\backslash 2^{j}B}\left|f\left(y\right)\right|\left|b\left(y\right)-b_B\right|{\left(\underset{B}{{\displaystyle \int }}{w}^q\left(x\right)\text{d}x\right)}^{\frac{1}{q}}\text{d}y}\\ \le C\frac{1}{r^{\frac{\left({\lambda }_1+{\lambda }_2\right)}{p_1}-\frac{{\lambda }_2}{q}+\frac{n}{p_2}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}\frac{1}{{\left|B\right|}^{1-\frac{\alpha }{n}}}\left[\underset{x\in B}{\sup }w\left(x\right)\right]\underset{j=1}{\stackrel{\infty }{{{\displaystyle \sum }}^{\text{}}}}{\displaystyle {\int }_{2^{j+1}B}\left|f\left(y\right)\right|\left|b\left(y\right)-b_B\right|\text{d}y{\left|B\right|}^{\frac{1}{t}}{\left|B\right|}^{\left(1-\frac{q}{t}\right)\frac{1}{q}}}\end{array}$

$\begin{array}{l}\le C\frac{1}{r^{\frac{\left({\lambda }_1+{\lambda }_2\right)}{p_1}-\frac{{\lambda }_2}{q}+\frac{n}{p_2}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}\frac{1}{{\left|B\right|}^{1-\frac{\alpha }{n}}}\left[\underset{x\in B}{\inf }w\left(x\right)\right]\underset{j=1}{\stackrel{\infty }{{{\displaystyle \sum }}^{\text{}}}}{\left|B\right|}^{\frac{1}{q}}{\displaystyle {\int }_{2^{j+1}B}\left|f\left(y\right)\right|\left|b\left(y\right)-b_{2^{j+1}B}\right|\text{d}y}\\ +C\frac{1}{r^{\frac{\left({\lambda }_1+{\lambda }_2\right)}{p_1}-\frac{{\lambda }_2}{q}+\frac{n}{p_2}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}\frac{1}{{\left|B\right|}^{1-\frac{\alpha }{n}}}\left[\underset{x\in B}{\inf }w\left(x\right)\right]\underset{j=1}{\stackrel{\infty }{{{\displaystyle \sum }}^{\text{}}}}{\left|B\right|}^{\frac{1}{q}}{\displaystyle {\int }_{2^{j+1}B}\left|f\left(y\right)\right|\left|b_{2^{j+1}B}-b_B\right|\text{d}y}\\ :=F_1+F_2。\end{array}$

对于 $F_1$ ，利用Hölder不等式，有

$\begin{array}{c}{F}_1\le C\frac{1}{r^{\frac{\left({\lambda }_1+{\lambda }_2\right)}{p_1}-\frac{{\lambda }_2}{q}+\frac{n}{p_2}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}\frac{1}{{\left|B\right|}^{1-\frac{\alpha }{n}}}\left[\underset{x\in B}{\inf }w\left(x\right)\right]\underset{j=1}{\stackrel{\infty }{{{\displaystyle \sum }}^{\text{}}}}{\left|B\right|}^{\frac{1}{q}}{\displaystyle {\int }_{2^{j+1}B}\left|f\left(y\right)\right|\left|b\left(y\right)-b_{2^{j+1}B}\right|\text{d}y}\\ \le C\frac{{\left|B\right|}^{\frac{1}{p_2}}}{r^{\frac{\left({\lambda }_1+{\lambda }_2\right)}{p_1}-\frac{{\lambda }_2}{q}+\frac{n}{p_2}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}\frac{1}{{\left|B\right|}^{1-\frac{\alpha }{n}+\frac{1}{p_2}}}\frac{w\left(B\right)}{\left|B\right|}\underset{j=1}{\stackrel{\infty }{{{\displaystyle \sum }}^{\text{}}}}{\left|B\right|}^{\frac{1}{q}}{\displaystyle {\int }_{2^{j+1}B}\left|f\left(y\right)\right|\left|b\left(y\right)-b_{2^{j+1}B}\right|\text{d}y}\\ \le C\frac{1}{r^{\frac{\left({\lambda }_1+{\lambda }_2\right)}{p_1}-\frac{{\lambda }_2}{q}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}\underset{j=1}{\stackrel{\infty }{{{\displaystyle \sum }}^{\text{}}}}\frac{{\left|B\right|}^{\frac{1}{q}}}{{\left|B\right|}^{1-\frac{\alpha }{n}+\frac{1}{p_2}}}{\Vert b\Vert }_{*}{\left|2^{j+1}B\right|}^{\frac{1}{p_2}}{\left({\displaystyle {\int }_{2^{j+1}B}{\left|f\left(y\right)\right|}^{p_1}{w}^{p_1}\left(y\right)\text{d}y}\right)}^{\frac{1}{p_1}}{\left|2^{j+1}B\right|}^{1-\frac{1}{p_1}-\frac{1}{p_2}}\end{array}$

$\begin{array}{l}\le C\underset{j=1}{\stackrel{\infty }{{{\displaystyle \sum }}^{\text{}}}}\frac{r^{\frac{{\lambda }_1}{p_1}}{w}^{p_1}{\left(2^{j+1}B\right)}^{\frac{{\lambda }_2}{n{p}_1}}}{r^{\frac{\left({\lambda }_1+{\lambda }_2\right)}{p_1}-\frac{{\lambda }_2}{q}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}{\Vert b\Vert }_{*}\frac{{\left|B\right|}^{\frac{1}{q}}}{{\left|B\right|}^{1-\frac{\alpha }{n}+\frac{1}{p_2}}}{\left(\frac{1}{r^{{\lambda }_1}{w}^{p_1}{\left(2^{j+1}B\right)}^{\frac{{\lambda }_2}{n}}}{\displaystyle {\int }_{2^{j+1}B}{\left|f\left(y\right)\right|}^{p_1}{w}^{p_1}\left(y\right)\text{d}y}\right)}^{\frac{1}{p_1}}{\left|2^{j+1}B\right|}^{1-\frac{1}{p_1}}\\ \le C{\Vert b\Vert }_{*}{\Vert f\Vert }_{L^{p_{1,\left({\lambda }_1,{\lambda }_2\right)}}\left(w^{p_1}\right)}\underset{j=1}{\stackrel{\infty }{{{\displaystyle \sum }}^{\text{}}}}\frac{w^q{\left(2^{j+1}B\right)}^{\frac{{\lambda }_2}{qn}}}{w^q{\left(B\right)}^{\frac{{\lambda }_2}{qn}}}\frac{{\left|B\right|}^{\frac{1}{q}}}{{\left|B\right|}^{1-\frac{\alpha }{n}+\frac{1}{p_2}}}{\left|2^{j+1}B\right|}^{1-\frac{1}{p_1}}\\ \le C{\Vert b\Vert }_{*}{\Vert f\Vert }_{L^{p_{1,\left({\lambda }_1,{\lambda }_2\right)}}\left(w^{p_1}\right)}\underset{j=1}{\stackrel{\infty }{{{\displaystyle \sum }}^{\text{}}}}{\left|2^{j+1}\right|}^{1-\frac{1}{p_1}+\frac{{\lambda }_2}{qn}}{\left|B\right|}^{\frac{1}{q}-\frac{1}{p_1}-\frac{1}{p_2}+\frac{\alpha }{n}}\\ \le C{\Vert b\Vert }_{*}{\Vert f\Vert }_{L^{p_{1,\left({\lambda }_1,{\lambda }_2\right)}}\left(w^{p_1}\right)}\underset{j=1}{\stackrel{\infty }{{{\displaystyle \sum }}^{\text{}}}}{\left|2^{j+1}\right|}^{1-\frac{1}{p_1}+\frac{{\lambda }_2}{qn}}.\end{array}$

根据 $1+\frac{{\lambda }_2}{qn}\le \frac{1}{p_1}$ ，结合引理3，有

$F_1\le C{\Vert b\Vert }_{*}{\Vert f\Vert }_{L^{p_{1,\left({\lambda }_1,{\lambda }_2\right)}}\left(w^{p_1}\right)}$ . (18)

对于 $F_2$ ，由于

$\begin{array}{c}\left|b_{2^{j+1}B}-b_B\right|\le \underset{k=0}{\stackrel{j}{{{\displaystyle \sum }}^{\text{}}}}\left|b_{2^{k+1}B}-b_{2^{k}B}\right|\\ \le C\underset{k=0}{\stackrel{j}{{{\displaystyle \sum }}^{\text{}}}}{\left(\frac{1}{\left|2^{k+1}B\right|}{\displaystyle {\int }_{2^{k+1}B}{\left|b\left(y\right)-b_{2^{k+1}B}\right|}^{p_2}\text{d}y}\right)}^{\frac{1}{p_2}}\\ \le C\left(j+1\right){\Vert b\Vert }_{*}.\end{array}$

因此，利用Hölder不等式，有

$\begin{array}{c}{F}_2\le C\frac{1}{r^{\frac{\left({\lambda }_1+{\lambda }_2\right)}{p_1}-\frac{{\lambda }_2}{q}+\frac{n}{p_2}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}\frac{1}{{\left|B\right|}^{1-\frac{\alpha }{n}}}\left[\underset{x\in B}{\inf }w\left(x\right)\right]\underset{j=1}{\stackrel{\infty }{{{\displaystyle \sum }}^{\text{}}}}{\left|B\right|}^{\frac{1}{q}}{\displaystyle {\int }_{2^{j+1}B}\left|f\left(y\right)\right|\left|b_{2^{j+1}B}-b_B\right|\text{d}y}\\ \le C\frac{1}{r^{\frac{\left({\lambda }_1+{\lambda }_2\right)}{p_1}-\frac{{\lambda }_2}{q}+\frac{n}{p_2}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}\frac{1}{{\left|B\right|}^{1-\frac{\alpha }{n}}}\frac{w\left(B\right)}{\left|B\right|}\underset{j=1}{\stackrel{\infty }{{{\displaystyle \sum }}^{\text{}}}}{\left|B\right|}^{\frac{1}{q}}\left|b_{2^{j+1}B}-b_B\right|{\displaystyle {\int }_{2^{j+1}B}\left|f\left(y\right)\right|\text{d}y}\\ \le C\frac{{\left|B\right|}^{\frac{1}{p_2}}}{r^{\frac{\left({\lambda }_1+{\lambda }_2\right)}{p_1}-\frac{{\lambda }_2}{q}+\frac{n}{p_2}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}\frac{{\left|B\right|}^{\frac{1}{q}}}{{\left|B\right|}^{1-\frac{\alpha }{n}+\frac{1}{p_2}}}\underset{j=1}{\stackrel{\infty }{{{\displaystyle \sum }}^{\text{}}}}\left(j+1\right){\Vert b\Vert }_{*}{\left({\displaystyle {\int }_{2^{j+1}B}{\left|f\left(y\right)\right|}^{p_1}{w}^{p_1}\left(y\right)\text{d}y}\right)}^{\frac{1}{p_1}}{\left|2^{j+1}B\right|}^{1-\frac{1}{p_1}}\end{array}$

$\begin{array}{l}\le C{\Vert b\Vert }_{*}{\Vert f\Vert }_{L^{p_{1,\left({\lambda }_1,{\lambda }_2\right)}}\left(w^{p_1}\right)}\underset{j=1}{\stackrel{\infty }{{{\displaystyle \sum }}^{\text{}}}}\left(j+1\right)\frac{{\left|B\right|}^{\frac{1}{q}}}{{\left|B\right|}^{1-\frac{\alpha }{n}+\frac{1}{p_2}}}\frac{r^{\frac{{\lambda }_1}{p_1}}{w}^{p_1}{\left(2^{j+1}B\right)}^{\frac{{\lambda }_2}{n{p}_1}}}{r^{\frac{\left({\lambda }_1+{\lambda }_2\right)}{p_1}-\frac{{\lambda }_2}{q}}{\left(w^q\left(B\right)\right)}^{\frac{{\lambda }_2}{qn}}}{\left|2^{j+1}B\right|}^{1-\frac{1}{p_1}}\\ \le C{\Vert b\Vert }_{*}{\Vert f\Vert }_{L^{p_{1,\left({\lambda }_1,{\lambda }_2\right)}}\left(w^{p_1}\right)}\underset{j=1}{\stackrel{\infty }{{{\displaystyle \sum }}^{\text{}}}}\left(j+1\right)\frac{{\left|B\right|}^{\frac{1}{q}}}{{\left|B\right|}^{1-\frac{\alpha }{n}+\frac{1}{p_2}}}\frac{w^q{\left(2^{j+1}B\right)}^{\frac{{\lambda }_2}{qn}}}{w^q{\left(B\right)}^{\frac{{\lambda }_2}{qn}}}{\left|2^{j+1}B\right|}^{1-\frac{1}{p_1}}\\ \le C{\Vert b\Vert }_{*}{\Vert f\Vert }_{L^{p_{1,\left({\lambda }_1,{\lambda }_2\right)}}\left(w^{p_1}\right)}\underset{j=1}{\stackrel{\infty }{{{\displaystyle \sum }}^{\text{}}}}\left(j+1\right){\left|2^{j+1}\right|}^{1-\frac{1}{p_1}+\frac{{\lambda }_2}{qn}}{\left|B\right|}^{\frac{1}{q}-\frac{1}{p_1}-\frac{1}{p_2}+\frac{\alpha }{n}}\\ \le C{\Vert b\Vert }_{*}{\Vert f\Vert }_{L^{p_{1,\left({\lambda }_1,{\lambda }_2\right)}}\left(w^{p_1}\right)}\underset{j=1}{\stackrel{\infty }{{{\displaystyle \sum }}^{\text{}}}}{\left|2^{j+1}\right|}^{1-\frac{1}{p_1}+\frac{{\lambda }_2}{qn}}.\end{array}$

根据 $1+\frac{{\lambda }_2}{qn}\le \frac{1}{p_1}$ ，利用引理3，有

$F_2\le C{\Vert b\Vert }_{*}{\Vert f\Vert }_{L^{p_{1,\left({\lambda }_1,{\lambda }_2\right)}}\left(w^{p_1}\right)}$ . (19)

结合式(15)~(19)，定理3证毕。