众所周知, 经典的多线性Calderón-Zygmund理论始于Coifman和Meyer的工作^[1-3]. 自Lacey和Thiele^[12-13]在双线性Hilbert变换取得突破性进展后, Grafakos和Torres在文献[9]和[10]中系统的研究了多线性Calderón-Zygmund理论.Kenig和Stein^[11]研究了多线性分数次积分算子. 此后, 越来越多的数学家投身于这一领域的研究并取得了丰硕的成果.

设$ m $是自然数, $ K(y_0, y_1, \cdots, y_m) $是定义在$ ({{\Bbb R}}^{n})^{m+1} $上远离对角线$ y_0 = y_1 = \cdots = y_m $的函数, 定义多线性算子$ T $为

(1.1) $ \begin{eqnarray} T(f_1, \cdots, f_m)(x) = \int_{R^{mn}}K(x, y_1, \cdots, y_m)\prod\limits_{j = 1}^mf_j(y_j){\rm d}y_1\cdots {\rm d}y_m, \end{eqnarray} $

其中$ f_j, j = 1, \cdots, m $是具有紧支撑集的函数且$ x\not\in\bigcap\limits_{j = 1}^m {\rm supp} f_j $.

特别地, 称$ K(x, y_1, \cdots, y_m) $为$ m $ -线性Calderón-Zygmund核, 如果它满足如下条件.

(ⅰ) 尺寸条件: 对于所有的$ ({{\Bbb R}}^n)^{m+1} $中远离对角线的点$ (y_0, y_1, \cdots, y_m) $, 存在$ C>0 $使得

(1.2) $ \begin{eqnarray} |K(y_0, y_1, \cdots, y_m)|\le\frac{C}{(\sum\limits_{k, \ell = 0}^m|y_k-y_{\ell}|)^{mn}}; \end{eqnarray} $

(ⅱ) 光滑条件: 存在$ \varepsilon>0 $, 当$ 0\le j\le m $且$ |y_j-y'_j|\le \frac12\max\limits_{0\le k\le m}\{|y_k-y_j|\} $, 有

(1.3) $ \begin{eqnarray} |K(y_0, y_1, \cdots, y_j, \cdots, y_m)-K(y_0, y_1, \cdots, y'_j, \cdots, y_m)|\le\frac{C|y_j-y'_j|^\varepsilon}{(\sum\limits_{k, \ell = 0}^m|y_k-y_{\ell}|)^{mn+\varepsilon}}. \end{eqnarray} $

称与$ m $ -线性Calderón-Zygmund核$ K $相关的多线性算子$ T $为多线性Calderón-Zygmund算子, 如果$ T $满足下列条件中之一

(ⅰ) 当$ r>1 $时, $ T $是$ L^{r_1, 1}\times\cdots\times L^{r_m, 1} $到$ L^{r, \infty} $上的有界算子;

(ⅱ) 当$ r = 1 $时, $ T $是$ L^{r_1, 1}\times\cdots\times L^{r_m, 1} $到$ L^{1} $上的有界算子.

近来, Lu和Zhang^[18]引进了$ \omega $型多线性Calderón-Zygmund核且得到了这类多线性奇异积分算子的有界性. 设$ \omega(t) $是定义在$ {{\Bbb R}} _+ $上的非负非减的函数.称定义在$ ({{\Bbb R}} ^n)^{m+1} $上远离对角线$ y_0 = y_1 = \cdots = y_m $的局部可积函数$ K(y_0, y_1, \cdots, y_m) $为$ \omega $型$ m $ -线性Calderón-Zygmund核, 如果它满足尺寸条件(1.2)和

(1.4) $ \begin{eqnarray} &&|K(y_0, \cdots, y_j, \cdots, y_m)-K(y_0, \cdots, y'_j, \cdots, y_m)|\\ &\le& \frac{C}{(|y_0-y_1|+\cdots+|y_0-y_m|)^{mn}}\omega\bigg(\frac{|y_j-y'_j|}{|y_0-y_1|+\cdots+|y_0-y_m|}\bigg), \end{eqnarray} $

其中$ 1\le j\le m $和$ |y_j-y'_j|\le \frac12\max\limits_{1\le j\le m}\{|y_0-y_j|\} $. 显然当$ \omega(t) = t^\varepsilon $时, $ \omega $ -型$ m $ -线性算子$ T $就是标准的多线性Calderón-Zygmund算子.

称$ K(y_0, \cdots, y_m) $为$ m $ -线性广义核, 如果$ K(y_0, \cdots, y_m) $满足条件(1.2)和如下积分条件: 对任意$ k_1, \cdots, k_m\in{\mathbb N}_{+} $, 存在$ C_{k_i}, i = 1, \cdots, m $, 使得

(1.5) $ \begin{eqnarray} &&\bigg(\int_{2^{k_m}|x-x'|\le|y_m-x|<2^{k_m+1}|x-x'|}\cdots\int_{2^{k_1}|x-x'|\le|y_1-x|<2^{k_1+1}|x-x'|}\\ &&|K(x, y_1, \cdots, y_m)-K(x', y_1, \cdots, y_m)|^q{\rm d}y_1\cdots {\rm d}y_m\bigg)^{\frac{1}{q}}\\ &\le& C|x-x'|^{-\frac{mn}{q'}}\prod\limits_{i = 1}^m C_{k_i}2^{-\frac{nk_i}{q'}}, \end{eqnarray} $

其中$ 1/q+1/q' = 1 $和$ 1<q<\infty $. 称核函数满足条件(1.2)和(1.5)的多线奇异积分算子$ T $为具有广义核的多线性Calderón-Zygmund算子. Lin在文献[14]得到了具有广义核的多线性Calderón-Zygmund算子在乘积加权Lebesgue空间上的有界性, 同时Lin在文献[14]中也指出当$ C_{k_i} = \omega(2^{-k_i})^{\frac1{m}} $时, (1.5)式就是(1.4)式.

设局部可积函数$ \vec{b} = (b_1, \cdots, b_m) $, 由多线性Calderón-Zygmund算子$ T $和函数$ \vec{b} $生成的$ m $ -线性交换子$ T_{\Sigma\vec{b}} $为

(1.6) $ \begin{equation} T_{\Sigma\vec{b}}(f_1, \cdots , f_m)(x) = \sum\limits_{j = 1}^mT_{b_j}^j(f_1, \cdots , f_m)(x), \end{equation} $

其中

$ T_{b_j}^j(f_1, \cdots , f_m)(x) = b_j(x)T(f_1, \cdots , f_m)(x)-T(f_1, \cdots , b_jf_j, \cdots , f_m)(x). $

Pérez和Torres在文献[19]中证明了当$ \vec{b}\in BMO^m $时, $ m $ -线性交换子$ T_{\Sigma\vec{b}} $是$ L^{p_1}(\omega)\times\cdots\times L^{p_m}(\omega) $到$ L^{p}(\omega) $上有界的, 其中$ 1<p_1, \cdots , p_m<\infty $和$ 1/p = 1/p_1+\cdots+1/p_m $.后来, Lerner等在文献[16]证明了当$ \vec{\omega}\in A_{\vec{P}} $时, $ m $ -线性交换子$ T_{\Sigma\vec{b}} $是$ L^{p_1}(\omega_1)\times\cdots\times L^{p_m}(\omega_m) $到$ L^p(\nu_{\vec{\omega}}) $上有界的, 其中$ 1<p_1, \cdots, p_m<\infty $且$ 1/p = 1/p_1+\cdots+1/p_m $. 2017年, Lin和Xiao证明了具有广义核的$ m $ -线性Calderón-Zygmund算子也是$ L^{p_1}(\omega_1)\times\cdots\times L^{p_m}(\omega_m) $到$ L^p(\nu_{\vec{\omega}}) $上有界的, 推广了文献[19]和[16]的结果.

20世纪80年代, Fabes, Jerison和Kenig研究了多线性Littlewood-Paley $ g $函数且发现多线性Littlewood-Paley型估计在偏微分和其他领域中具有重要的应用^[5-7]. 此后涌现了越来越多的有关多线性Littlewood-Paley型算子的有界性结果^{[17, 21-22]}. Xue和Yan^[21]给出了如下的核定义.

定义1.1  CZ$ I $ -型积分光滑条件

(ⅰ) 设$ K_t(x, y_1, \cdot, y_m) = t^{-mn}K(\frac{x}{t}, \frac{y_1}{t}, \cdots, \frac{y_m}{t}) $是定义在$ ({{\Bbb R}} ^{n})^{m+1} $上远离对角线$ x = y_1 = \cdots = y_m $的局部可积函数, 存在常数$ C>0 $使得

(1.7) $ \begin{equation} \bigg(\int_0^\infty|K_t(x, y_1, \cdots, y_m)|^2\frac{{\rm d}t}{\rm t}\bigg)^{\frac{1}{2}} \le\frac{A}{(\sum\limits_{k = 1}^m|x-y_k|)^{mn}}, \end{equation} $

(1.8) $ \begin{equation} \bigg(\int_0^\infty|K_t(x, y_1, \cdots, y_m)-K_t(z, y_1, \cdots, y_m)|^2\frac{{\rm d}t}{\rm t}\bigg)^{\frac{1}{2}} \le\frac{A|x-z|^{\varepsilon}}{(\sum\limits_{k = 1}^m|x-y_k|)^{mn+\varepsilon}}, \end{equation} $

其中$ |x-z|\le\frac12\max\limits_{j = 1}^m\{|x-y_j|\} $;

(ⅱ)

(1.9) $ \begin{equation} \bigg(\int_0^\infty|K_t(x, y_1, \cdots, y_m)-K_t(z, y_1, \cdots, y_m)|^2\frac{{\rm d}t}{\rm t}\bigg)^{\frac{1}{2}} \le\frac{A|y_j-y'_j|^{\varepsilon}}{(\sum\limits_{k = 1}^m|x-y_k|)^{mn+\varepsilon}}, \end{equation} $

其中$ |y_j-y'_j|\le\frac12\max\limits_{j = 1}^m\{|x-y_j|\} $.

定义与$ K $相关的多线性平方算子$ T $为

(1.10) $ \begin{equation} T(\vec{f})(x) = \bigg(\int_0^\infty|\int_{{{\Bbb R}} ^{nm}}K_t(x, y_1, \cdots, y_m)f_1(y_1)\cdots f_m(y_m){\rm d}\vec{y}|^2\frac{{\rm d}t}{\rm t}\bigg)^{\frac12}, \end{equation} $

其中$ \vec{f} = (f_1, \cdots, f_m)\in (S({{\Bbb R}} ^n))^m $和$ x\not\in\bigcap\limits_{j = 1}^m{\rm supp} f_j $. Xue和Yan证明了多线性平方算子$ T $是$ L^{p_1}(\omega_1)\times\cdots\times L^{p_m}(\omega_m) $到$ L^{p}(\nu_{\omega}) $上有界, 其中$ \nu_{\omega} = \prod\limits_{i = 1}^m\omega_i^{\frac{p}{p_i}} $且$ \frac{1}{p_1}+\cdots+\frac{1}{p_m} = \frac1{p} $. 随后, Si和薛^[20]改进了文献[21]中的结果, 他们多形性平方算子$ T $的核函数的由CZ$ I $ -型积分光滑条件核降低为一类$ \omega $ - 型积分核.

定义1.2  $ \omega $ -型积分条件

(ⅰ) 设$ K_t(x, y_1, \cdot, y_m) = t^{-mn}K(\frac{x}{t}, \frac{y_1}{t}, \cdots, \frac{y_m}{t}) $是定义在$ ({{\Bbb R}} ^{n})^{m+1} $上远离对角线$ x = y_1 = \cdots = y_m $的局部可积函数, 存在常数$ C>0 $使得

(1.11) $ \begin{equation} \bigg(\int_0^\infty|K_t(x, y_1, \cdots, y_m)|^2\frac{{\rm d}t}{\rm t}\bigg)^{\frac{1}{2}} \le\frac{A}{(\sum\limits_{k = 1}^m|x-y_k|)^{mn}}, \end{equation} $

(1.12) $ \begin{equation} \bigg(\int_0^\infty|K_t(x, y_1, \cdots, y_m)-K_t(z, y_1, \cdots, y_m)|^2\frac{{\rm d}t}{\rm t}\bigg)^{\frac{1}{2}} \le \frac{A}{(\sum\limits_{k = 1}^m|x-y_k|)^{mn}}\omega\bigg(\frac{|x-z|}{(\sum\limits_{k = 1}^m|x-y_k|)}\bigg), \end{equation} $

其中$ |x-z|\le\frac12\max\limits_{j = 1}^m\{|x-y_j|\} $;

(ⅱ)

(1.13) $ \begin{equation} \bigg(\int_0^\infty|K_t(x, y_1, \cdots, y_m)-K_t(z, y_1, \cdots, y_m)|^2\frac{{\rm d}t}{\rm t}\bigg)^{\frac{1}{2}} \le\frac{A}{(\sum\limits_{k = 1}^m|x-y_k|)^{mn}}\omega\bigg(\frac{|y_j-y'_j|}{(\sum\limits_{k = 1}^m|x-y_k|)}\bigg), \end{equation} $

其中$ |y_j-y'_j|\le\frac12\max\limits_{j = 1}^m\{|x-y_j|\} $.

Si和xue证明了当$ \omega(t)\in Dini(1) $时, 具有$ \omega(t) $ -型积分核的多线性平方算子也是$ L^{p_1}(\omega_1)\times\cdots\times L^{p_m}(\omega_m) $到$ L^{p}(\nu_{\omega}) $上有界.下面我们介绍一类广义积分核条件:

定义1.3  (H$ _1) $: 设$ K_t(x, y_1, \cdot, y_m) = t^{-mn}K(\frac{x}{t}, \frac{y_1}{t}, \cdots, \frac{y_m}{t}) $是定义在$ (R^{n})^{m+1} $上远离对角线$ x = y_1 = \cdots = y_m $的局部可积函数, 存在常数$ C>0 $使得

(1.14) $ \begin{eqnarray} \bigg(\int_0^\infty|K_t(x, y_1, \cdots, y_m)|^2\frac{{\rm d}t}{\rm t}\bigg)^{\frac{1}{2}} \le\frac{A}{(\sum\limits_{k = 1}^m|x-y_k|)^{mn}}, \end{eqnarray} $

(H$ _2): $设$ k_1, \cdots, k_m\in{\mathbb N}_{+} $, 存在$ C_{k_i}(i = 1, \cdots, m) $, 使得

(1.15) $ \begin{eqnarray} &&\bigg(\int_{2^{k_m}|x-x'|\le|y_m-x|<2^{k_m+1}|x-x'|}\cdots\int_{2^{k_1}|x-x'|\le|y_1-x|<2^{k_1+1}|x-x'|}\\ &&\bigg(\int_0^\infty|K_t(x, y_1, \cdots, y_m)-K_t(x', y_1, \cdots, y_m)|^2\frac{{\rm d}t}{\rm t}\bigg)^{\frac{q}{2}}{\rm d}y_1\cdots {\rm d}y_m\bigg)^{\frac{1}{q}}\\ &\le & C|x-x'|^{-\frac{mn}{q'}}\prod\limits_{i = 1}^mC_{k_i}2^{-\frac{nk_i}{q'}}, \end{eqnarray} $

其中$ 1/q+1/q' = 1 $和$ 1<q<\infty $.称满足条件(H$ _1) $和(H$ _2) $的$ K_t $为广义积分核.称广义积分核$ K_t $相关的多线性平方算子$ T $为具有广义核的多线性平方算子.

显然, 当$ C_{k_i} = \omega(2^{k_i})^{\frac1{m}} $, 广义积分核函数满足Si和Xue定义的$ \omega $型积分条件.受文献[15, 21]和[20]的启发, 我们将研究具有广义积分核的多线性平方算子在乘积Lebesgue空间上的加权有界性及其交换子的加权有界性.

我们首先得到了多线性平方算子的加权Lebesgue空间上的乘积估计.

定理1.1  设$ m\geq2, T $是定义(1.10)式中的多线性平方算子.对于$ i = 1, \cdots, m $, 核满足条件(H$ _{1}) $和(H$ _{2}) $且$ \sum\limits_{k_i = 1}^\infty C_{k_i}<\infty $.假设$ 1\le r_1, \cdots, r_m\le q' $且$ \frac{1}{r} = \frac{1}{r_1}+\cdots+\frac{1}{r_m} $和$ T $是从$ L^{r_1}\times \cdots\times L^{r_m} $连续映射到$ L^{r, \infty} $.则对于任意$ q'<p_1, \cdots, p_m<\infty $且$ \frac{1}{p} = \frac{1}{p_1}+\cdots+\frac{1}{p_m} $, 则$ T $从$ L^{p_1}(\omega_1)\times\cdots\times L^{p_m}(\omega_m) $连续映射到$ L^{p}(\omega) $. 其中$ (\omega_1, \cdots, \omega_m)\in (A_{p_1/q'}, \cdots, A_{p_m/q'}) $, $ \omega = \prod\limits_{j = 1}^m\omega_j^{\frac{p}{p_j}} $.

接下来研究的是多线性平方算子$ T $的$ (L^\infty\times\cdots\times L^\infty, BMO) $有界性.

定理1.2  设$ m\geq2, T $是(1.10)式中的多线性平方算子.对于$ i = 1, \cdots, m $, 核满足条件(H$ _{1}) $和(H$ _{2}) $且$ \sum\limits_{k_i = 1}^\infty C_{k_i}<\infty $. 假设$ 1\le r_1, \cdots, r_m\le q' $且$ \frac{1}{r} = \frac{1}{r_1}+\cdots+\frac{1}{r_m} $和$ T $是从$ L^{r_1}\times \cdots\times L^{r_m} $连续映射到$ L^{r, \infty} $.则$ T $是从$ L^{\infty}\times\cdots\times L^{\infty} $连续映射到$ BMO $.

下面定理讨论的是平方算子的交换子的加权范数不等式.

定理1.3  设$ m\geq2, T $是(1.6)式中的多线性平方算子的交换子.对于$ i = 1, \cdots, m $, 核满足条件(H$ _{1}) $和(H$ _{2}) $且$ \sum\limits_{k_i = 1}^\infty C_{k_i}<\infty $.假设$ 1\le r_1, \cdots, r_m\le q' $且$ \frac{1}{r} = \frac{1}{r_1}+\cdots+\frac{1}{r_m} $和$ T $是从$ L^{r_1}\times \cdots\times L^{r_m} $连续映射到$ L^{r, \infty} $.若$ \vec{b}\in BMO^m $, 对于任意$ q'<p_1, \cdots, p_m<\infty $且$ \frac{1}{p} = \frac{1}{p_1}+\cdots+\frac{1}{p_m} $, $ T_{\sum \vec{b}} $从$ L^{p_1}(\omega_1)\times\cdots\times L^{p_m}(\omega_m) $连续映射到$ L^{p}(\omega) $.其中$ (\omega_1, \cdots, \omega_m)\in (A_{p_1/q'}, \cdots, A_{p_m/q'}) $, $ \omega = \prod\limits_{j = 1}^m\omega_j^{\frac{p}{p_j}} $.

注1.1   根据前面核函数的讨论, 定理1.1改进了文献[21]和[20]中的强型加权估计.事实上, 定理1.3也改进核函数条件为CZ$ I $ -型积分光滑条件的多线性平方函数的$ m $ -线性交换子在乘积Lebesgue空间的加权估计结论.

首先介绍一簇Muckenhoupt权$ \omega\in A_p $且$ 1<p<\infty $. 设$ 1<p<\infty $, 如果对任意$ B\subset {{\Bbb R}} ^n $, 存在常数$ C $使得

$ \left(\frac{1}{|Q|}\int_Q\omega(x){\rm d}x\right)\left(\frac{1}{|Q|}\int_Q\omega(x)^{1-p^{\prime}}{\rm d}x\right)^{p-1}\leq C<\infty, $

则称权函数$ \omega\in A_p $; 如果存在常数$ C $, 使得

$ \frac{1}{|Q|}\int_Q\omega(x){\rm d}x\leq C\omega(x), \ \ {\rm a.e.}\ x\in {{\Bbb R}} ^n, $

则称权函数$ \omega\in A_1 $; 并且定义$ A_\infty = \bigcup\limits_{1\leq p<\infty}A_p. $更多权函数的性质参见文献[8].

现在回顾经典的Hardy-Littlewood中心极大函数$ M $和sharp极大函数$ M^{\sharp} $的定义

$ M(f)(x) = \sup\limits_{Q\ni x}\frac{1}{|Q|}\int_Q|f(y)|{\rm d}y $

和

$ M^\sharp f(x) = \sup\limits_{Q\ni x}\inf\limits_c\frac{1}{|Q|}\int_Q|f(y)-C|{\rm d}y = \sup\limits_{Q\ni x}\frac{1}{|Q|}\int_Q|f(y)-f_Q|{\rm d}y, $

其中$ f_Q = \frac{1}{|Q|}\int_Q f(y){\rm d}y $, $ Q $为$ {{\Bbb R}} ^n $中的球.

接下给出另一些极大算子的定义, 对$ \delta>0 $, 定义

$ {\cal M}_\delta(f)(x) = \left(M(|f|^\delta)\right)^{1/\delta}(x) = \bigg(\sup\limits_{B\ni x}\frac{1}{|B|}\int_B|f(y)|^r{\rm d}y\bigg) ^{1/\delta} $

和

$ M_r^\sharp(f)(x) = \left(M^\sharp(|f|^r)\right)^{1/r}(x). $

根据文献[8]可知, $ M $是$ L^p(\omega) $有界当且仅当$ \omega\in A_p. $$ {\cal M}_r $在$ L^p(\omega) $有界当且仅当$ \omega\in A_p. $

引理2.1(Kolmogorov不等式)^[8]  设$ 0<p<q<\infty, $存在常数$ C = C_{p, q} $, 使得对任意的可测函数$ f $, 有

$ \big\|f\big\|_{L^p(B, \frac{{\rm d}x}{|B|})}\leq C\big\|f\big\|_{L^{q, \infty}(B, \frac{{\rm d}x}{|B|})}. $

引理2.2^[8]  (ⅰ) 对$ 1\leq p<q\leq\infty, A_p\subset A_q; $

(ⅱ) 若$ \omega\in A_1, $对$ 0\leq\theta\leq1, $有$ \omega^\theta\in A_1; $

(ⅲ) 对$ 1<p<\infty, \omega\in A_p $当且仅当$ \omega^{1-p^{\prime}}\in A_{p^{\prime}}. $

下面的引理是由Fefferman和Stein在文献[4]所得到的.

引理2.3^[4]  设$ 0<p, \delta<\infty $且$ \omega\in A_\infty({{\Bbb R}} ^n). $则存在常数$ C>0, $使得

$ \int_{{{\Bbb R}} ^n}M_\delta f(x)^p\omega(x){\rm d}x\leq C\int_{{{\Bbb R}} ^n}M_\delta^\sharp f(x)^p\omega(x){\rm d}x, $

对任意的光滑函数$ f $, 不等式左边部分是有限的.

引理2.4^[8]  对$ (\omega_1, \cdots, \omega_m)\in (A_{p_1}, \cdots, A_{p_m}) $, $ 1\le p_1, \cdots, p_m<\infty $和$ 0<\theta_1, \cdots, \theta_m<1 $, 使得$ \theta_1+\cdots+\theta_m = 1 $, 有$ \omega_1^{\theta_1}\cdots\omega_m^{\theta_m}\in A_{\max\{p_1, \cdots, p_m\}} $.

在证明带广义核的多线性平方算子在加权Lebesgue空间上的有界性之前，下面先介绍多线性平方算子的sharp极大函数的估计.

引理3.1  设$ m\geq2, T $是定义(1.10)中的多线性平方算子.对于$ i = 1, \cdots, m $, 核满足条件(H$ _1) $和(H$ _2) $且$ \sum\limits_{k_i = 1}^\infty C_{k_i}<\infty $.假设$ 1\le r_1, \cdots, r_m\le q' $且$ \frac{1}{r} = \frac{1}{r_1}+\cdots+\frac{1}{r_m} $和$ T $是从$ L^{r_1}\times \cdots\times L^{r_m} $到$ L^{r, \infty} $的连续映射.若$ 0< \delta < \min \{1, \frac{q}{m}\} $, 有

$ \begin{eqnarray*} M_{\delta}^\sharp(T(\vec{f}))(x)\le C\prod\limits_{j = 1}^mM_{q'}(f_j)(x). \end{eqnarray*} $

证  取以$ x_Q $为中心和边长为$ r_{Q} $的方体$ Q = Q(x_Q, r_{Q}) $. 设$ \lambda_i = (b_i)_{Q^*}, i = 1, 2 $. 将$ f_i $做如下分解

$ f_i = f_i\chi_{Q^*}+f_i\chi_{{{\Bbb R}} ^n\backslash Q^*} = f_i^0+f_i^\infty, $

其中$ Q^\ast = 16Q. $对于$ 0 <\delta< \frac{q}m $，选取$ z_0\in 3Q\backslash 2Q $, 有

$ \begin{eqnarray*} &&\bigg(\frac1{|Q|}\int_{Q}\big||T(\vec{f})(z)|^{\delta}-|T(f_1^\infty, \cdots, f_m^\infty)(z_0)|^{\delta}|{\rm d}z\bigg)^{\frac1{\delta}}\\ &\le& C\left(\frac1{|Q|}\int_{Q}\big|T(\vec{f})(z)-T(f_1^\infty, \cdots, f_m^\infty)(z_0)\big|^{\delta}{\rm d}z\right)^{\frac1{\delta}}\\ &\le& C\left(\frac1{|Q|}\int_{Q}|T(f_1^0, \cdots, f_m^0)(z)|{\rm d}z\right)+C\sum\limits_{\vec{\alpha}\ne 0}\left(\frac1{|Q|}\int_{Q}|T(f_1^{\alpha_1}, f_2^{\alpha_2}, \cdots, f_m^{\alpha_m})(z)|^{\delta}{\rm d}z\right)^{\frac1{\delta}}\\ & &+C\left(\frac1{|Q|}\int_{Q}|T(f_1^\infty, f_2^\infty, \cdots, f_m^\infty)(z)-T(f_1^\infty, \cdots, f_m^\infty)(z_0)|^{\delta}{\rm d}z\right)^{\frac1{\delta}}\\ & = &I+II+III, \end{eqnarray*} $

其中$ \vec{\alpha} = (\alpha_1, \cdots, \alpha_m) $和$ (\alpha_1, \cdots, \alpha_m)\ne (\infty, \cdots, \infty) $.

由引理2.1和多线性平方算子$ T $在$ (L^{r_1}\times\cdots\times L^{r_m}, L^{r, \infty}) $上的有界性, 可得

$ \begin{eqnarray*} I\le C\|T(f_1^0, \cdots, f_m^0)\|_{L^{r, \infty}}\le C\prod\limits_{j = 1}^m\bigg(\frac{1}{|16Q|}\int_{16Q}|f_j(y_j)|^{r_j}{\rm d}y_j\bigg)^{\frac{1}{r_{j}}}\le C\prod\limits_{j = 1}^mM_{q'}(f_j)(x). \end{eqnarray*} $

先估计$ II $.假设$ \alpha_1 = \cdots = \alpha_l = \infty $和$ \alpha_{l+1} = \cdots = \alpha_{\infty} = 0 $. 根据Hölder不等式, Minkowski不等式得

$ \begin{eqnarray*} II&\le& C\sum\limits_{\vec{\alpha}\ne 0}\frac{1}{|Q|}\int_{Q}|T(f_1^{\alpha_1}, f_2^{\alpha_2}, \cdots, f_m^{\alpha_m})(z)|{\rm d}z\\ &\le &C\sum\limits_{\vec{\alpha}\ne 0}\frac{1}{|Q|}\int_{Q}\int_{{{\Bbb R}} ^{nm}}\bigg(\int_0^\infty|K_v(z, y_1, \cdots, y_m)|^2\frac{dv}{v}\bigg)^{\frac{1}{2}}\prod\limits_{j = 1}^m|f_j^{\alpha_j}(y_j)|{\rm d}y_1\cdots {\rm d}y_m{\rm d}z\\ &\le &C\frac{1}{|Q|}\int_{Q}\bigg(\int_{(16Q)^{m-l}}\prod\limits_{j = l+1}^m|f_j(y_j)|{\rm d}y_{l+1}\cdots {\rm d}y_m\int_{({{\Bbb R}} ^n\backslash 16Q)^{l}} \frac{\prod\limits_{j = 1}^l|f_j(y_j)|}{|z-y_j|^{mn}}{\rm d}y_1\cdots {\rm d}y_l\bigg){\rm d}z\\ &\le &C\prod\limits_{i = l+1}^{m}\bigg(\frac{1}{|16Q|}\int_{16Q}|f_i(y_i)|{\rm d}y_i\bigg)\prod\limits_{j = 1}^l\bigg(\sum\limits_{k = 4}^\infty 2^{-kn}\frac{1}{|2^kQ|} \int_{2^{k+1}Q\backslash 2^kQ}|f_j(y_j)|{\rm d}y_j\bigg)\\ &\le& C\prod\limits_{j = 1}^{m}M(f_i)(x)\le C\prod\limits_{j = 1}^{m}M_{q'}(f_i)(x). \end{eqnarray*} $

对于$ z\in Q $和$ y_1, \cdots, y_m\in (16Q)^c $, 有$ |y_j-z_0|\ge 2|z-z_0| $和$ r_B\le|z-z_0|\le 4r_B $.运用Hölder不等式, Minkowski不等式得

$ \begin{eqnarray*} III&\le &C\left(\frac{1}{|Q|}\int_{Q}|T(f_1^\infty, f_2^\infty, \cdots, f_m^\infty)(z)-T(f_1^\infty, f_2^\infty, \cdots, f_m^\infty)(z_0)|{\rm d}z\right)\\ &\le&\frac{C}{|Q|}\int_{Q}\bigg(\sum\limits_{k_1 = 1}^{\infty}\cdots\sum\limits_{k_m = 1}^{\infty}\int_{2^{k_m}|z-z_0|\le|y_1-z_0|<2^{k_m+1}|z-z_0|}\cdots\int_{2^{k_1}|z-z_0|\le|y_1-z_0|<2^{k_1+1}|z-z_0|}\\ &&\bigg(\int_0^\infty|K_v(z, y_1, \cdots, y_m)-K_v(z_0, y_1, \cdots, y_m)|^2\frac{{\rm d}v}{v}\bigg)^{\frac{1}{2}}\prod\limits_{j = 1}^m|f_j(y_j)|{\rm d}y_1\cdots {\rm d}y_m\bigg){\rm d}z\\ &\le&\frac{C}{|Q|}\int_{Q}\bigg(\sum\limits_{k_1 = 1}^{\infty}\cdots\sum\limits_{k_m = 1}^{\infty}\int_{2^{k_m}|z-z_0|\le|y_1-z_0|<2^{k_m+1}|z-z_0|}\cdots\int_{2^{k_2}|z-z_0|\le|y_1-z_0|<2^{k_2+1}|z-z_0|}\\ &&\bigg(\int_{2^{k_1}|z-z_0|\le|y_1-z_0|<2^{k_1+1}|z-z_0|}\bigg(\int_0^\infty|K_v(z, y_1, \cdots, y_m)\\ & &-K_v(z_0, y_1, \cdots, y_m)|^2\frac{{\rm d}v}{v}\bigg)^{\frac{q}{2}}{\rm d}y_1\bigg)^{\frac{1}{q}}\bigg(\int_{2^{k_1+4}Q} |f_1(y_1)|^{q'}{\rm d}y_1\bigg)^{\frac{1}{q'}}\prod\limits_{j = 2}^m|f_j(y_j)|{\rm d}y_2\cdots {\rm d}y_m\bigg){\rm d}z\\ &\le&\frac{C}{|Q|}\int_{Q}\bigg(\sum\limits_{k_1 = 1}^{\infty}\cdots\sum\limits_{k_m = 1}^{\infty}\bigg(\int_{2^{k_m}|z-z_0|\le|y_1-z_0|<2^{k_m+1}|z-z_0|}\cdots\int_{2^{k_2}|z-z_0|\le|y_1-z_0|<2^{k_2+1}|z-z_0|}\\ &&\int_{2^{k_1}|z-z_0|\le|y_1-z_0|<2^{k_1+1}|z-z_0|}\bigg(\int_0^\infty|K_v(z, y_1, \cdots, y_m)\\ &&-K_v(z_0, y_1, \cdots, y_m)|^2\frac{{\rm d}v}{v}\bigg)^{\frac{q}{2}}{\rm d}y_1 {\rm d}y_2\cdots {\rm d}y_m\bigg)^{\frac{1}{q}} \prod\limits_{j = 1}^m\bigg(\int_{2^{k_j+4}Q}|f_j(y_j)|^{q'}{\rm d}y_j\bigg)^{\frac{1}{q'}}\bigg){\rm d}z\\ &\le&\frac{C}{|Q|}\int_{Q}\bigg(\sum\limits_{k_1 = 1}^{\infty}\cdots\sum\limits_{k_m = 1}^{\infty} \prod\limits_{j = 1}^m\bigg(\frac{1}{|2^{k_j+4}Q|}\int_{2^{k_j+4}Q}|f_j(y_j)|^{q'}{\rm d}y_j\bigg)^{\frac{1}{q'}}\\ &&\prod\limits_{j = 1}^m|2^{k_j+4}Q|^{\frac{1}{q'}}|z-z_0|^{-\frac{mn}{q'}}\prod\limits_{j = 1}^mC_{k_j}2^{-\frac{k_jn}{q'}}\bigg){\rm d}z\\ &\le& C\prod\limits_{j = 1}^mM_{q'}(f_j)(x)(\sum\limits_{k_1 = 1}^\infty C_{k_1})\times\cdots\times(\sum\limits_{k_m = 1}^\infty C_{k_m})\le C\prod\limits_{j = 1}^mM_{q'}(f_j)(x). \end{eqnarray*} $

最后结合$ I, II $和$ III $的估计, 则引理证毕.

下面利用多线性平方算子的sharp极大函数估计证明定理1.1.

定理1.1的证明  由引理2.4知$ \omega\in A_{\max\{p_1/q, \cdots, p_m/q\}}\subset A_{\infty} $. 选取$ 0<\delta<\frac1{m} $, 利用引理2.3和引理3.1得

$ \begin{eqnarray*} \|T(\vec{f})\|_{L^p(\omega)}&\le &\|M_{\delta}(T(\vec{f}))\|_{L^p(\omega)}\le \|M_{\delta}^{\sharp}(T(\vec{f}))\|_{L^p(\omega)}\\ &\le &C\|\prod\limits_{j = 1}^mM_{q'}(f_j)(x)\|_{L^{p}(\omega)}\le C\prod\limits_{j = 1}^m\|M_{q'}(f_j)(x)\|_{L^{p_j}(\omega_j)}\le C\|f_j\|_{L^{p_j}(\omega_j)}. \end{eqnarray*} $

证明完毕.

定理1.2的证明  设$ f_1, \cdots, f_m\in L^\infty $.对任意球$ B(x_0, r) $, 将$ f_j $分解为

$ \begin{eqnarray*} f_j = f_j\chi_{2B}+f_j\chi_{{{\Bbb R}} ^n\backslash 2Q} = f_j^0+f_j^\infty, \quad j = 1, 2, \cdots, m, \end{eqnarray*} $

则可得

$ \begin{eqnarray*} &&\frac{1}{|B|}\int_{B}|T(f_1, \cdots, f_m)(z)-T(f_1^\infty, \cdots, f_m^\infty)(z_0)|{\rm d}z\\ &\le& \frac{1}{|B|}\int_{B}|T(f_1^0, \cdots, f_m^0)(z)|{\rm d}z+\sum\limits_{\vec{\alpha}\ne 0, \vec{\alpha}\ne (\infty, \cdots, \infty)} \frac{1}{|B|}\int_{B}|T(f_1^{\alpha_1}, \cdots, f_m^{\alpha_m})(z)|{\rm d}z\\ &&+\frac{1}{|B|}\int_{B}|T(f_1^\infty, \cdots, f_m^\infty)(z)-T(f_1^\infty, \cdots, f_m^\infty)(z_0)|{\rm d}z\\ &: = &D_1+D_2+D_3. \end{eqnarray*} $

利用Hölder不等式和多线性平方算子$ T $在$ (L^{p_1}\times \cdots\times L^{p_m}, L^p) $上的有界性得

$ \begin{eqnarray*} D_1&\le &\bigg(\frac{1}{|B|}\int_B|T(f_1^0, \cdots, f_m^0)(z)|^p{\rm d}z\bigg)^{\frac{1}{p}}\le C |B|^{-\frac{1}{p}}\|f_1^0\|_{L^{p_1}}\|f_2^0\|_{L^{p_2}}\cdots\|f_m^0\|_{L^{p_m}}\\ &\le &C\prod\limits_{j = 1}^m\|f_j\|_{L^\infty}|B|^{-\frac{1}{p}}|2B|^{\frac{1}{p_1}+\cdots+\frac{1}{p_m}}\le C\prod\limits_{j = 1}^m\|f_j\|_{L^\infty}. \end{eqnarray*} $

现在估计$ D_2 $. 假设$ \alpha_1 = \cdots = \alpha_l = \infty $和$ \alpha_{l+1} = \cdots = \alpha_{\infty} = 0 $.利用Hölder不等式, Minkowski不等式得

$ \begin{eqnarray*} D_2&\le &C\sum\limits_{\vec{\alpha}\ne 0}\frac{1}{|B|}\int_{B}|T(f_1^{\alpha_1}, f_2^{\alpha_2}, \cdots, f_m^{\alpha_m})(z)|{\rm d}z\\ &\le& C\sum\limits_{\vec{\alpha}\ne 0}\frac{1}{|B|}\int_{Q}\int_{{{\Bbb R}} ^{nm}}\bigg(\int_0^\infty|K_v(z, y_1, \cdots, y_m)|^2\frac{{\rm d}v}{v}\bigg)^{\frac{1}{2}}\prod\limits_{j = 1}^m|f_j^{\alpha_j}(y_j)|{\rm d}y_1\cdots {\rm d}y_m{\rm d}z\\ &\le &C\frac{1}{|B|}\int_{B}\bigg(\int_{(2B)^{m-l}}\prod\limits_{j = l+1}^m|f_j(y_j)|{\rm d}y_{l+1}\cdots {\rm d}y_m\int_{({{\Bbb R}} ^n\backslash 2B)^{l}} \frac{\prod\limits_{j = 1}^l|f_j(y_j)|}{|z-y_j|^{mn}}{\rm d}y_1\cdots {\rm d}y_l\bigg){\rm d}z\\ &\le &C\prod\limits_{j = 1}^m\|f_j\|_{L^\infty}\frac{1}{|Q|}\int_{Q}\bigg(\int_{(2B)^{m-l}}{\rm d}y_{l+1}\cdots {\rm d}y_m\int_{({{\Bbb R}} ^n\backslash 2B)^{l}} \frac{1}{|z-y_j|^{mn}}{\rm d}y_1\cdots {\rm d}y_l\bigg){\rm d}z\\ &\le &C\prod\limits_{j = 1}^m\|f_j\|_{L^\infty}. \end{eqnarray*} $

对于$ z\in B $和$ y_1, \cdots, y_m\in (2Q)^c $, 有$ |y_j-z_0|\ge 2|z-z_0| $. 根据Minkowski不等式, 则有

$ \begin{eqnarray*} D_3&\le &C\left(\frac{1}{|Q|}\int_{Q}|T(f_1^\infty, f_2^\infty, \cdots, f_m^\infty)(z)-T(f_1^\infty, f_2^\infty, \cdots, f_m^\infty)(z_0)|{\rm d}z\right)\\ &\le&\frac{C}{|Q|}\int_{Q}\bigg(\sum\limits_{k_1 = 1}^{\infty}\cdots\sum\limits_{k_m = 1}^{\infty}\int_{2^{k_m}|z-z_0|\le|y_1-z_0|<2^{k_m+1}|z-z_0|}\cdots\int_{2^{k_1}|z-z_0|\le|y_1-z_0|<2^{k_1+1}|z-z_0|}\\ &&\bigg(\int_0^\infty|K_v(z, y_1, \cdots, y_m)-K_v(z_0, y_1, \cdots, y_m)|^2\frac{{\rm d}v}{v}\bigg)^{\frac{1}{2}}\prod\limits_{j = 1}^m|f_j(y_j)|{\rm d}y_1\cdots {\rm d}y_m\bigg){\rm d}z\\ &\le&\frac{C}{|Q|}\int_{Q}\bigg(\sum\limits_{k_1 = 1}^{\infty}\cdots\sum\limits_{k_m = 1}^{\infty}\int_{2^{k_m}|z-z_0|\le|y_1-z_0|<2^{k_m+1}|z-z_0|}\cdots\int_{2^{k_2}|z-z_0|\le|y_1-z_0|<2^{k_2+1}|z-z_0|}\\ &&\bigg(\int_{2^{k_1}|z-z_0|\le|y_1-z_0|<2^{k_1+1}|z-z_0|}\bigg(\int_0^\infty|K_v(z, y_1, \cdots, y_m)\\ &&-K_v(z_0, y_1, \cdots, y_m)|^2\frac{{\rm d}v}{v}\bigg)^{\frac{q}{2}}{\rm d}y_1\bigg)^{\frac{1}{q}} \bigg(\int_{2^{k_1+4}Q}|f_1(y_1)|^{q'}{\rm d}y_1\bigg)^{\frac{1}{q'}}\prod\limits_{j = 2}^m|f_j(y_j)|{\rm d}y_2\cdots {\rm d}y_m\bigg){\rm d}z\\ &\le&\frac{C}{|Q|}\int_{Q}\bigg(\sum\limits_{k_1 = 1}^{\infty}\cdots\sum\limits_{k_m = 1}^{\infty}\bigg(\int_{2^{k_m}|z-z_0|\le|y_1-z_0|<2^{k_m+1}|z-z_0|}\cdots\int_{2^{k_2}|z-z_0|\le|y_1-z_0|<2^{k_2+1}|z-z_0|}\\ &&\int_{2^{k_1}|z-z_0|\le|y_1-z_0|<2^{k_1+1}|z-z_0|}\bigg(\int_0^\infty|K_v(z, y_1, \cdots, y_m)\\ &&-K_v(z_0, y_1, \cdots, y_m)|^2\frac{{\rm d}v}{v}\bigg)^{\frac{q}{2}}{\rm d}y_1 {\rm d}y_2\cdots {\rm d}y_m\bigg)^{\frac{1}{q}} \prod\limits_{j = 1}^m\bigg(\int_{2^{k_j+4}Q}|f_j(y_j)|^{q'}{\rm d}y_j\bigg)^{\frac{1}{q'}}\bigg){\rm d}z\\ &\le&\frac{C}{|Q|}\int_{Q}\bigg(\sum\limits_{k_1 = 1}^{\infty}\cdots\sum\limits_{k_m = 1}^{\infty} \prod\limits_{j = 1}^m\bigg(\frac{1}{|2^{k_j+4}Q|}\int_{2^{k_j+4}Q}|f_j(y_j)|^{q'}{\rm d}y_j\bigg)^{\frac{1}{q'}}\\ &&\prod\limits_{j = 1}^m|2^{k_j+4}Q|^{\frac{1}{q'}}|z-z_0|^{-\frac{mn}{q'}}\prod\limits_{j = 1}^mC_{k_j}2^{-\frac{k_jn}{q'}}\bigg){\rm d}z\\ &\le& C\prod\limits_{j = 1}^m\|f_j\|_{L^\infty}(\sum\limits_{k_1 = 1}^\infty C_{k_1})\times\cdots\times(\sum\limits_{k_m = 1}^\infty C_{k_m})\le C\prod\limits_{j = 1}^m\|f_j\|_{L^\infty}. \end{eqnarray*} $

定理1.2证明完毕.

在证明定理1.3之前, 下面先介绍多线性平方算子交换子的中心极大函数的估计.

引理5.1  设$ m\geq2, T $是定义(1.10)中的多线性平方算子.对于$ i = 1, \cdots, m $, 核满足条件(H$ _1) $和(H$ _2) $且$ \sum\limits_{k_i = 1}^\infty C_{k_i}<\infty $. 假设$ 1\le r_1, \cdots, r_m\le q' $且$ \frac{1}{r} = \frac{1}{r_1}+\cdots+\frac{1}{r_m} $和$ T $是从$ L^{r_1}\times \cdots\times L^{r_m} $连续映射到$ L^{r, \infty} $.若$ 0<\delta<\min\{1, \frac{q}{m}\}, \delta<\epsilon<\infty, \vec{b}\in BMO^m $和$ q'<s<\infty $, 则对所有紧支集可测函数$ \vec{f} = (f_1, \cdots, f_m) $, 有

$ \begin{eqnarray*} M_{\delta}^\sharp(T_{\sum \vec{b}}(\vec{f}))(x)\le C\|\vec{b}\|_{BMO}\bigg(M_\epsilon(T(\vec{f}))(x)+\prod\limits_{j = 1}^mM_{s}(f_j)(x)\bigg). \end{eqnarray*} $

证  固定$ x_Q $为中心和边长为$ r_{Q} $的方体$ Q = Q(x_Q, r_{Q}) $.为了简化计算, 仅考虑$ m = 2 $的情形.令$ \lambda_i = (b_i)_{Q^*} $, 将$ f_i $分解成为

$ f_i = f_i\chi_{Q^*}+f_i\chi_{{{\Bbb R}} ^n\backslash Q^*} = f_i^0+f_i^\infty, $

其中$ Q^\ast = 16Q. $选取$ z_0\in 3Q\backslash 2Q $, 有

$ \begin{eqnarray*} T_{b}(\vec{f})(z)& = &(b(z)-b_{16Q})T(\vec{f})(z)-T((b-b_{16Q}f_1, \cdots, f_m))(z)\\ & = &(b(z)-b_{16Q})T(\vec{f})(z)-T((b-b_{16Q})f_1^0, \cdots, f_m^0))(z)\\ &&-\sum\limits_{\vec{\alpha}\ne 0, \vec{\alpha}\ne(\infty, \cdots, \infty)}T((b-b_{16Q})f_1^{\alpha_1}, f_2^{\alpha_2}, \cdots, f_m^{\alpha_m})) -T((b-b_{16Q})f_1^\infty, \cdots, f_m^\infty))(z). \end{eqnarray*} $

则

$ \begin{eqnarray*} &&\bigg(\frac{1}{|Q|}\int_{Q}\big|T_{b}(\vec{f})(z)-T((b-b_{16Q})f_1^\infty, \cdots, f_m^\infty))(z_0)\big|^{\delta}{\rm d}z\bigg)^{\frac{1}{\delta}}\\ &\le& C\left(\frac{1}{|Q|}\int_{Q}|(b(z)-b_{16Q})T(\vec{f})(z)|^{\delta}{\rm d}z\right)^{\frac{1}{\delta}}\\ &&+C\left(\frac{1}{|Q|}\int_{Q}|T((b-b_{16Q})f_1^0, \cdots, f_m^0))(z)|^{\delta}{\rm d}z\right)^{\frac{1}{\delta}}\\ &&+C\sum\limits_{\vec{\alpha}\ne 0, \vec{\alpha}\ne(\infty, \cdots, \infty)}\left(\frac{1}{|Q|}\int_{Q}|T((b-b_{16Q})f_1^{\alpha_1}, f_2^{\alpha_2}, \cdots, f_m^{\alpha_m})(z)|^{\delta}{\rm d}z\right)^{\frac{1}{\delta}}\\ &&+C\left(\frac{1}{|Q|}\int_{Q}|T(f_1^\infty, f_2^\infty, \cdots, f_m^\infty)(z)-T((b-b_{16Q})f_1^\infty, \cdots, f_m^\infty)(z_0)|^{\delta}{\rm d}z\right)^{\frac{1}{\delta}}\\ & = &I+II+III+IV, \end{eqnarray*} $

其中$ \vec{\alpha} = (\alpha_1, \cdots, \alpha_m) $.由于$ 0<\delta<\frac{1}{m} $和$ \delta<\epsilon<\infty $, 存在$ 1<t<\min\{\frac{\epsilon}{\delta}, \frac{\delta}{1-\delta}\} $.则有$ \delta t<\epsilon $和$ \delta t'>1 $.利用Hölder不等式可得

$ \begin{eqnarray*} I&\le& C\bigg(\frac{1}{|Q|}\int_{Q}|(b(z)-b_{16Q})|^{\delta t'}{\rm d}z\bigg)^{\frac{1}{\delta t'}}\bigg(\frac{1}{|Q|}\int_{Q}|T(\vec{f})(z)|^{\delta t}{\rm d}z\bigg)^{\frac{1}{\delta t}}\\ &\le& C\|b\|_{BMO}\bigg(\frac{1}{|Q|}\int_{Q}|T(\vec{f})(z)|^{\epsilon}{\rm d}z\bigg)^{\frac{1}{\epsilon}}\\ &\le& C\|b\|_{BMO}M_{\epsilon}(T(\vec{f}))(x). \end{eqnarray*} $

由于$ q'<s<\infty $, 记$ u = \frac{s}{q'} $, 则$ 1<t<\infty $. 根据$ 0<\delta<r<\infty $和引理 得

$ \begin{eqnarray*} II&\le& C\|T((b-b_{16Q})f_1^0, \cdots, f_m^0)(z)\|_{L^\delta(Q, \frac{{\rm d}z}{|Q|})}\le C\|T((b-b_{16Q})f_1^0, \cdots, f_m^0)(z)\|_{L^{r, \infty}(Q, \frac{{\rm d}z}{|Q|})}\\ &\le& C\bigg(\frac{1}{|Q|}\int_{16Q}|b(y_1)-b_{16Q}|^{r_1}|f_1(y_1)|^{r_1}{\rm d}y_1\bigg)^{\frac{1}{r_1}}\prod\limits_{j = 2}^m\bigg(\frac{1}{|Q|}\int_{16Q}|f_j(y_j)|^{r_j}{\rm d}y_1\bigg)^{\frac{1}{r_j}}\\ &\le &C\bigg(\frac{1}{|Q|}\int_{16Q}|b(y_1)-b_{16Q}|^{u'r_1}{\rm d}y_1\bigg)^{\frac{1}{u'r_1}}\bigg(\frac{1}{|Q|}\int_{16Q}|f_1(y_1)|^{ur_1}{\rm d}y_1\bigg)^{\frac{1}{ur_1}}\prod\limits_{j = 2}^mM_{r_j}(f_j)(x)\\ &\le &C\|b\|_{BMO}\prod\limits_{j = 1}^mM_{s}(f_j)(x). \end{eqnarray*} $

先估计$ II $, 假设$ \alpha_1 = \cdots = \alpha_l = \infty $和$ \alpha_{l+1} = \cdots = \alpha_{\infty} = 0 $.运用Hölder不等式, Minkowski不等式得

$ \begin{eqnarray*} II&\le& C\sum\limits_{\vec{\alpha}\ne 0}\frac{1}{|Q|}\int_{Q}|T((b-b_{16Q})f_1^{\alpha_1}, f_2^{\alpha_2}, \cdots, f_m^{\alpha_m})(z)|{\rm d}z\\ &\le& C\sum\limits_{\vec{\alpha}\ne 0}\frac{1}{|Q|}\int_{Q}\int_{{{\Bbb R}} ^{nm}}\bigg(\int_0^\infty|K_v(z, y_1, \cdots, y_m)|^2\frac{{\rm d}v}{v}\bigg)^{\frac{1}{2}}\\ &&\times|(b-b_{16Q})f_1^{\alpha_1}|\prod\limits_{j = 2}^m|f_j^{\alpha_j}(y_j)|{\rm d}y_1\cdots {\rm d}y_m{\rm d}z\\ &\le& C\frac{1}{|Q|}\int_{Q}\bigg(\int_{(16Q)^{m-l}}\prod\limits_{j = l+1}^m|f_j(y_j)|{\rm d}y_{l+1}\cdots {\rm d}y_m\\ &&\times\int_{({{\Bbb R}} ^n\backslash 16Q)^{l}} \frac{|(b-b_{16Q})f_1^{\alpha_1}|\prod\limits_{j = 2}^l|f_j(y_j)|}{|z-y_j|^{mn}}{\rm d}y_1\cdots {\rm d}y_l\bigg){\rm d}z\\ &\le& C\prod\limits_{i = l+1}^{m}\bigg(\frac{1}{|16Q|}\int_{16Q}|f_i(y_i)|{\rm d}y_i\bigg)\bigg(\sum\limits_{k = 4}^\infty 2^{-kn}\frac{1}{|2^kQ|} \int_{2^{k+1}Q\backslash 2^kQ}|b(y_1)-b_{16Q}||f_1(y_1)|{\rm d}y_1\bigg)\\ &&\times\prod\limits_{j = 2}^l\bigg(\sum\limits_{k = 4}^\infty 2^{-kn}\frac{1}{|2^kQ|} \int_{2^{k+1}Q\backslash 2^kQ}|f_j(y_j)|{\rm d}y_j\bigg)\\ &\le& C\|b\|_{BMO}M_s(f_1)(x)\prod\limits_{j = 1}^{m}M(f_i)(x)\le C\|b\|_{BMO}\prod\limits_{j = 1}^{m}M_{s}(f_i)(x). \end{eqnarray*} $

对于$ z\in Q $和$ y_1, \cdots, y_m\in (16Q)^c $, 则有$ |y_j-z_0|\ge 2|z-z_0| $和$ r_Q\le|z-z_0|\le 4r_Q $. 记$ \frac{1}{q}+\frac{1}{uq'}+\frac{1}{u'q'} = 1 $, 利用Hölder不等式, Minkowski不等式可得

$ \begin{eqnarray*} III&\le& C\bigg\{\frac{1}{|Q|}\int_{Q}|T((b-b_{16Q})f_1^\infty, f_2^\infty, \cdots, f_m^\infty)(z)-T((b-b_{16Q})f_1^\infty, f_2^\infty, \cdots, f_m^\infty)(z_0)|{\rm d}z\bigg\}\\ &\le&\frac{C}{|Q|}\int_{Q}\bigg(\sum\limits_{k_1 = 1}^{\infty}\cdots\sum\limits_{k_m = 1}^{\infty}\int_{2^{k_m}|z-z_0|\le|y_1-z_0|<2^{k_m+1}|z-z_0|}\cdots\int_{2^{k_1}|z-z_0|\le|y_1-z_0|<2^{k_1+1}|z-z_0|} \\ &&\bigg(\int_0^\infty|K_v(z, y_1, \cdots, y_m)-K_v(z_0, y_1, \cdots, y_m)|^2\frac{{\rm d}v}{v}\bigg)^{\frac{1}{2}}\\ &&\times|(b(y_1)-b_{16Q})||f_1(y_1)|\prod\limits_{j = 2}^m|f_j(y_j)|{\rm d}y_1\cdots {\rm d}y_m\bigg){\rm d}z\\ &\le&\frac{C}{|Q|}\int_{Q}\bigg(\sum\limits_{k_1 = 1}^{\infty}\cdots\sum\limits_{k_m = 1}^{\infty}\int_{2^{k_m}|z-z_0|\le|y_1-z_0|<2^{k_m+1}|z-z_0|}\cdots\int_{2^{k_2}|z-z_0|\le|y_1-z_0|<2^{k_2+1}|z-z_0|}\\ &&\bigg(\int_{2^{k_1}|z-z_0|\le|y_1-z_0|<2^{k_1+1}|z-z_0|}\bigg(\int_0^\infty|K_v(z, y_1, \cdots, y_m)\\ &&-K_v(z_0, y_1, \cdots, y_m)|^2\frac{{\rm d}v}{v}\bigg)^{\frac{q}{2}}{\rm d}y_1\bigg)^{\frac{1}{q}}\bigg(\int_{2^{k_1+4}Q}|b(y_1)-b_{16Q}|^{uq'}{\rm d}y_1\bigg)^{\frac{1}{uq'}}\\ & &\bigg(\int_{2^{k_1+4}Q}|f_1(y_1)|^{u'q'}{\rm d}y_1\bigg)^{\frac{1}{u'q'}}\prod\limits_{j = 2}^m|f_j(y_j)|{\rm d}y_2\cdots {\rm d}y_m\bigg){\rm d}z\\ &\le&\frac{C}{|Q|}\int_{Q}\bigg(\sum\limits_{k_1 = 1}^{\infty}\cdots\sum\limits_{k_m = 1}^{\infty}\bigg(\int_{2^{k_m}|z-z_0|\le|y_1-z_0|<2^{k_m+1}|z-z_0|}\cdots\int_{2^{k_2}|z-z_0|\le|y_1-z_0|<2^{k_2+1}|z-z_0|}\\ & &\int_{2^{k_1}|z-z_0|\le|y_1-z_0|<2^{k_1+1}|z-z_0|}\bigg(\int_0^\infty|K_v(z, y_1, \cdots, y_m)\\ &&-K_v(z_0, y_1, \cdots, y_m)|^2\frac{{\rm d}v}{v}\bigg)^{\frac{q}{2}}{\rm d}\vec{y}\bigg)^{\frac{1}{q}} \bigg(\int_{2^{k_1+4}Q}|b(y_1)-b_{16Q}|^{uq'}{\rm d}y_1\bigg)^{\frac{1}{uq'}}\\ &&\bigg(\int_{2^{k_1+4}Q}|f_1(y_1)|^{u'q'}{\rm d}y_1\bigg)^{\frac{1}{u'q'}}\prod\limits_{j = 2}^m\bigg(\int_{2^{k_j+4}Q}|f_j(y_j)|^{q'}{\rm d}y_j\bigg)^{\frac{1}{q'}}\bigg){\rm d}z\\ &\le&\frac{C}{|Q|}\int_{Q}\bigg(\sum\limits_{k_1 = 1}^{\infty}\cdots\sum\limits_{k_m = 1}^{\infty}\bigg(\frac{1}{|2^{k_1+4}Q|}\int_{2^{k_1+4}Q}|b(y_1)-b_{16Q}|^{uq'}{\rm d}y_1\bigg)^{\frac{1}{uq'}}\\ &&\bigg(\frac{1}{|2^{k_1+4}Q|}\int_{2^{k_1+4}Q}|f_1(y_1)|^{u'q'}{\rm d}y_1\bigg)^{\frac{1}{u'q'}} \prod\limits_{j = 2}^m\bigg(\frac{1}{|2^{k_j+4}Q|}\int_{2^{k_j+4}Q}|f_j(y_j)|^{q'}{\rm d}y_j\bigg)^{\frac{1}{q'}}\\ &&\prod\limits_{j = 1}^m|2^{k_j+4}Q|^{\frac1{q'}}|z-z_0|^{-\frac{mn}{q'}}\prod\limits_{j = 1}^mC_{k_j}2^{-\frac{k_jn}{q'}}\bigg){\rm d}z\\ &\le & C\|b\|_{BMO}\prod\limits_{j = 1}^mM_{q'}(f_j)(x)(\sum\limits_{k_1 = 1}^\infty k_1C_{k_1})\times\cdots\times(\sum\limits_{k_m = 1}^\infty C_{k_m})\\ &\le& C\prod\limits_{j = 1}^mM_{q'}(f_j)(x). \end{eqnarray*} $

结合$ I, II $和$ III $的估计, 引理证毕.

最后可以利用多线性平方算子交换子的sharp极大函数的点态估计证明定理1.3.

定理1.3的证明  选取$ \delta $和$ \epsilon $满足$ 0<\delta<\epsilon<\frac{1}{m} $. 利用引理5.1和引理2.3, 可得

$ \begin{eqnarray*} \|T_{\sum b}(\vec{f})\|_{L^p(\omega)}&\le& \|M_{\delta}(T_{\sum b}(\vec{f}))\|_{L^p(\omega)}\le \|M_{\delta}^{\sharp}(T_{\sum b}(\vec{f}))\|_{L^p(\omega)}\\ &\le& C\|\vec{b}\|_{BMO^m}\bigg(\|M_\epsilon(T(\vec{f}))\|_{L^p(\omega)}+\|\prod\limits_{j = 1}^mM_s(f_j)(x)\|_{L^p(\omega)}\bigg)\\ &\le &C\|\vec{b}\|_{BMO^m}\bigg(\|M(T(\vec{f}))\|_{L^p(\omega)}+\|\prod\limits_{j = 1}^mM_s(f_j)(x)\|_{L^p(\omega)}\bigg)\\ &\le &C\|\vec{b}\|_{BMO^m}\bigg(\|T(\vec{f})\|_{L^p(\omega)}+\|\prod\limits_{j = 1}^mM_s(f_j)(x)\|_{L^p(\omega)}\bigg)\\ &\le &C\|\vec{b}\|_{BMO^m}\bigg(\prod\limits_{j = 1}^m\|f_j\|_{L^{p_j}(\omega_j)}+\prod\limits_{j = 1}^m\|M_s(f_j)\|_{L^{p_j}(\omega_j)}\bigg)\\ &\le &C\|\vec{b}\|_{BMO^m}\prod\limits_{j = 1}^m\|f_j\|_{L^{p_j}(\omega_j)}, \end{eqnarray*} $

证明完毕.