设$0<l<n, $分数次积分$I_l$定义为

$ I_l f(x)=\int_{\mathbf{R}^n}\frac{f(y)}{|x-y|^{n-l}}dy. $

考虑分数次积分$I_l$与局部可积函数$b$生成的交换子$\ {[b, I_l]}f(x) = \ b(x)I_l f(x) -I_l(bf)(x).$当${1}/{q}={1}/{p}-{l}/{n}, 1<p<{n}/{l}$时, Chanillo在文献[1]中证明了$[b, I_l]$是$L^p$到$L^q$有界的充分必要条件是$b\in$ BMO.当$b$属于加权Lipschitz空间$ {\hbox{Lip}}_{\beta}(\mu) (0<\beta<1)$时, 陈爱清等在文献[2]中研究了$[b, I_l]$在加权Hardy空间上的有界性质.当$b$属于Lipschitz空间${\rm Lip}_{\beta}(0<\beta<1)$时, 陆善镇等在文献[3]中研究了$[b, I_l]$在Herz型Hardy空间上的有界性质, 他们的主要结果是

设$0<l<n-\beta, 1<q_1, q_2<\infty, 1/q_2=1/q_1-(l+\beta)/n.$

(ⅰ)若$0<p<\infty, n(1-1/q_1)\leq \alpha<n(1-1/q_1)+\beta, $则$[b, I_l]$是从$H\dot{K}^{\alpha, p}_{q_1}$到$\dot{K}^{\alpha, p}_{q_2}$的有界算子;

(ⅱ)若$0<p\leq 1, \alpha=n(1-1/q_1)+\beta$, 则$[b, I_l]$是从$H\dot{K}^{\alpha, p}_{q_1}$到$W\dot{K}^{\alpha, p}_{q_2}$的有界算子.

本文的目的是研究当$b$属于加权Lipschitz函数空间时, $[b, I_l]$在加权Herz型Hardy空间上的有界性质, 得到了如下结果:

定理1.1  设$\mu\in A_1, b\in {\hbox{Lip}}_{\beta}(\mu)(0<\beta<1), 0<l<n-\beta$.若$0<p<\infty, 1<q_1, q_2<\infty, 1/q_2=1/q_1-(l+\beta)/n$, 则当$n(1-1/q_1)\leq \alpha<n(1-1/q_1)+\beta$时, $[b, I_l]$是从$H\dot{K}^{\alpha, p}_{q_1}(\mu, \mu)$到$\dot{K}^{\alpha, p}_{q_2}(\mu, \mu^{1-(1-{l}/{n})q_{_2}})$的有界算子.

定理1.2  设$\mu\in A_1, b\in {\hbox{Lip}}_{\beta}(\mu)(0<\beta<1), 0<l<n-\beta$.若$0<p\leq 1, 1<q_1, q_2<\infty, 1/q_2=1/q_1-(l+\beta)/n$, 则当$\alpha=n(1-1/q_1)+\beta$时, $[b, I_l]$是从$H\dot{K}^{\alpha, p}_{q_1}(\mu, \mu)$到$W\dot{K}^{\alpha, p}_{q_2}(\mu, \mu^{1-(1-{l}/{n})q_{_2}})$的有界算子.

称定义在$\mathbf{R}^n$上的局部可积函数$\mu$属于Muckenhoupt $A_1$权, 如果

$ M(\mu)(x)\leq C\mu(x)\quad {\hbox{a.e.}}\ x\in \mathbf{R}^n. $

其中$M$表示标准的Hardy-Littlewood极大算子.

设$k\in \mathbb{Z}$, 记$B_k=B(0, 2^k), C_k=B_k\backslash B_{k-1}, \chi_k=\chi_{_{C_k}}.$ $A_1$权函数具有下面性质

引理2.1   如果$\mu\in A_1, $则存在常数$C$以及$0<\delta<1$, 使得当$k<j$时,

$ \frac{\mu(B_k)}{\mu(B_j)}\leq C2^{(k-j)n\delta}; $

当$k>j$时,

$ \frac{\mu(B_k)}{\mu(B_j)}\leq C2^{(k-j)n}. $

下面介绍加权Herz空间和加权Herz型Hardy空间的概念, 这些概念可参见文献[4-6].

定义2.1  设$\alpha\in \mathbf{R}, 0<p, q<\infty, \mu_1, \mu_2$为权函数, 齐次加权Herz空间定义为

$ \dot{K}^{\alpha, p}_q(\mu_1, \mu_2)=\left\{f\in L^q_{{\hbox {loc}}}(\mathbf{R}^n\backslash\{0\}, \mu_2):\|f\|_{\dot{K}^{\alpha, p}_q(\mu_1, \mu_2)}<\infty\right\}, $

其中

$ \|f\|_{\dot{K}^{\alpha, p}_q(\mu_1, \mu_2)}=\left\{\sum\limits_{j=-\infty}^{\infty}\mu_1(B_j)^{\frac{\alpha p}{n}}\|f\chi_{_{C_j}}\|^p_{L^q_{\mu_2}}\right\}^{\frac{1}{p}}. $

定义2.2   设$\alpha\in \mathbf{R}, 0<p, q<\infty, \mu_1, \mu_2$为权函数, 记$m_{j, \mu_2}(\lambda, f) =\mu_2(\{x\in C_j: |f(x)|>\lambda\}).$称$\mu_2$可测函数$f$属于齐次加权弱Herz空间, 如果

$ \|f\|_{W\dot{K}^{\alpha, p}_q(\mu_1, \mu_2)}=\sup\limits_{\lambda>0} \left\{\sum\limits_{j=-\infty}^{\infty}\mu_1(B_j)^{\frac{\alpha p}{n}}m_{j, \mu_2}(\lambda, f)^{\frac{p}{q}} \right\}^{\frac{1}{p}}<\infty. $

定义2.3  设$\alpha\in \mathbf{R}, 0<p, q<\infty, \mu_1, \mu_2$为权函数, 齐次加权Herz型Hardy空间定义为$H\dot{K}^{\alpha, p}_q(\mu_1, \mu_2)=\{f\in \mathcal{S}^{\prime}(\mathbf{R}^n): G(f)\in \dot{K}^{\alpha, p}_q(\mu_1, \mu_2)\}$且$\|f\|_{H\dot{K}^{\alpha, p}_q(\mu_1, \mu_2)} =\|G(f)\|_{\dot{K}^{\alpha, p}_q(\mu_1, \mu_2)} $, 其中$G(f)$为$f$的Grand极大函数.

当$\mu_1=\mu_2=1$时, 上述空间分别对应为齐次Herz空间, 齐次弱Herz空间以及齐次Herz型Hardy空间.

加权Herz型Hardy空间最重要的性质是中心原子分解.

定义2.4  设$\alpha\in \mathbf{R}, 1<q<\infty.$ $\mathbf{R}^n$上的函数$a$称为$(\alpha, q;\mu_1, \mu_2)$ -原子, 如果

(1) 存在$r>0, $使得supp$a\subset B(0, r);$

(2) $\|a\|_{L^q_{\mu_2}}\leq\mu_1(B)^{-\frac{\alpha}{n}};$

(3) 当$|\gamma|\leq [\alpha-n(1-1/q)]$时$\int_{R^n}a(x)x^\gamma dx=0.$

引理2.2^[6]  设$0<p<\infty, 1<q<\infty, \alpha\geq n(1-1/q), \mu_1, \mu_2\in A_1. \ \mathbf{R}^n$上的分布函数$f$属于$H\dot{K}^{\alpha, p}_q(\mu_1, \mu_2)$当且仅当存在支集为$B_k$的中心$(\alpha, q;\mu_1, \mu_2)$ -原子$a_k$和常数$\lambda_k, \sum\limits_{k=-\infty}^{\infty}|\lambda_k|^p<\infty, $使得$f=\sum\limits_{k=-\infty}^{\infty}\lambda_ka_k$在分布意义下成立, 并且

$\|f\|^p_{H\dot{K}^{\alpha, p}_q(\mu_1, \mu_2)}\sim \inf\left(\sum\limits_{k=-\infty}^\infty|\lambda_k|^p\right), $

其中下确界取自$f$的所有中心原子分解.

最后介绍加权Lipschitz空间及其性质.

定义2.5  设$\mu$为一个权函数, $1\leq p<\infty$, 一个局部可积函数函数$b$属于加权Lipschitz空间, 记为$b\in {\hbox{Lip}}_{\beta, p}(\mu)$, 如果

$ \sup\limits_B\frac{1}{\mu(B)^{{\beta}/{n}}}\left[\frac{1}{\mu(B)}\int_B|b(x)-b_B|^p \mu(x)^{1-p}dx\right]^{{1}/{p}}\leq C<\infty, $

这里$b_B=|B|^{-1}\int_B b(x)dx$, 上确界取遍所有的$B\subset \mathbf{R}^n$.上式中$C$的最小下界称为$b$的Lip$_{\beta, p}(\mu)$范数, 记为$\|b\|_{{\hbox{Lip}}_{_{\beta, p}}(\mu)}$.

当$\beta=0$时, ${\hbox{Lip}}_{_{\beta, p}}(\mu)$即为加权BMO空间; 当$\mu=1$时, Lip$_{_{\beta, p}}(\mu)$即为通常的Lipschitz空间.由文献[7], 如果$\mu\in A_1$, 则对任意的$1\leq p\leq q<\infty$有$ Lip_{\beta, p}(\mu)= {\hbox{Lip}}_{\beta, q}(\mu)$, 并且其范数等价.由于在下文中所涉及的权函数均属于$ A_1$, 为方便起见, 我们总记$ {\hbox{Lip}}_{\beta, p}(\mu)$为$ {\hbox{Lip}}_\beta(\mu)$, 其范数记为$\|\cdot\|_{{\hbox{Lip}}_{_\beta}(\mu)}$.加权Lipschitz函数有如下性质.

引理2.3^{[8, 9]}  设$1<q_{_1}, q_{_2}<\infty, 0<\beta<1, 0<l<n-\beta$且$1/q_{_2}=1/q_{_1}-(l+\beta)/n.$若$\mu\in A_1, $则$[b, I_l]$是$L^{q_{_1}}(\mu)$到$L^{q_2}(\mu^{1-(1-{l}/{n})q_{_2}})$有界的充分必要条件是$b\in {\hbox{Lip}}_{\beta}(\mu).$

引理2.4^[2]  若$\mu\in A_1$, $b\in {\hbox{Lip}}_\beta(\mu), j>k$, 则$|{b_{{B_j}}} - {b_{{B_k}}}| \le C{\left\| b \right\|_{{\rm{Li}}{{\rm{p}}_\beta }(\mu )}}{2^{j\beta - k(n + \beta )}}\mu {({B_k})^{1 + \frac{\beta }{n}}}.$

由引理2.2, $f$可以分解为$f=\sum\limits_{k=-\infty}^{\infty}\lambda_ka_k$, 其中$a_k$为支集是$B_k$的中心$(\alpha, q_{_1};\mu, \mu)$ -原子, 并且$\sum\limits_{k=-\infty}^{\infty}|\lambda_k|^p<\infty.$所以

$ \begin{array}{l} \left\| {[b,{I_l}]f} \right\|_{\dot K_{{q_2}}^{\alpha ,p}\left( {\mu ,{\mu ^{1 - \left( {1 - l/{n}} \right){q_2}}}} \right)}^p\\ \;\;\; \le C\sum\limits_{j = - \infty }^\infty \mu {({B_j})^{\frac{{\alpha p}}{n}}}{\left( {\sum\limits_{k = - \infty }^{j - 3} | {\lambda _k}{{\left\| {\left( {[b,{I_l}]{a_k}} \right){\chi _j}} \right\|}_{{L^{{q_2}}}({\mu ^{1 - (1 - l/n){q_2}}}}}} \right)^p}\\ \;\; + C\sum\limits_{j = - \infty }^\infty \mu {({B_j})^{\frac{{\alpha p}}{n}}}{\left( {\sum\limits_{k = j - 2}^\infty | {\lambda _k}|{{\left\| {\left( {[b,{I_l}]{a_k}} \right){\chi _j}} \right\|}_{{L^{{q_2}}}}}_{({\mu ^{1 - (1 - l/n){q_2}}})}} \right)^p}\\ = \;\;{M_1} + {M_2}, \end{array} $

由$[b, I_l]$的$L^{q_{_1}}(\mu)$到$L^{q_2}(\mu^{1-(1-{l}/{n})q_{_2}})$有界性,

$ \begin{array}{l} {M_2} \le C{\left\| b \right\|_{{\rm{Li}}{{\rm{p}}_\beta }(\mu )}}{\left( {\sum\limits_{j = - \infty }^\infty \mu {{({B_j})}^{\frac{{\alpha p}}{n}}}\sum\limits_{k = j - 2}^\infty | {\lambda _k}|{{\left\| {{a_k}} \right\|}_{{L^{{q_{_1}}}}(\mu )}}} \right)^p}\\ \;\;\;\;\; \le C{\left\| b \right\|_{{\rm{Li}}{{\rm{p}}_\beta }(\mu )}}\sum\limits_{j = - \infty }^\infty \mu {({B_j})^{\frac{{\alpha p}}{n}}}{\left( {\sum\limits_{k = j - 2}^\infty | {\lambda _k}|\mu {{({B_k})}^{ - \frac{\alpha }{n}}}} \right)^p}. \end{array} $

当$0<p\leq 1$时, 由引理2.1,

$ {M_2} \le C\left\| b \right\|{\rm{Li}}{{\rm{p}}_\beta }(\mu )\sum\limits_{k = - \infty }^\infty | {\lambda _k}{|^p}\sum\limits_{j = - \infty }^{k + 2} {\frac{{\mu ({B_j})}}{{\mu ({B_k})}}} \le C\left\| b \right\|{\rm{Li}}{{\rm{p}}_\beta }(\mu )\sum\limits_{k = - \infty }^\infty | {\lambda _k}{|^p}. $

当$p>1$时, 由Hölder不等式以及引理2.1,

$ \begin{array}{l} {M_2} \le C{\left\| b \right\|_{{\rm{Li}}{{\rm{p}}_\beta }(\mu )}}\sum\limits_{j = - \infty }^\infty {\left[ {\sum\limits_{k = j - 2}^\infty | {\lambda _k}{|^p}{{\left( {\frac{{\mu ({B_j})}}{{\mu ({B_k})}}} \right)}^{\frac{{\alpha p}}{{2n}}}}} \right]} {\left[ {\sum\limits_{k = j - 2}^\infty {{{\left( {\frac{{\mu ({B_j})}}{{\mu ({B_k})}}} \right)}^{\frac{{\alpha {p^\prime }}}{{2n}}}}} } \right]^{ - \frac{p}{{{p^\prime }}}}}\\ \;\;\;\;\;\; \le C{\left\| b \right\|_{{\rm{Li}}{{\rm{p}}_\beta }(\mu )}}\sum\limits_{k = - \infty }^\infty | {\lambda _k}{|^p}\sum\limits_{j = - \infty }^{k + 2} {{{\left( {\frac{{\mu ({B_j})}}{{\mu ({B_k})}}} \right)}^{\frac{{\alpha p}}{{2n}}}}} \le C{\left\| b \right\|_{{\rm{Li}}{{\rm{p}}_\beta }(\mu )}}\sum\limits_{k = - \infty }^\infty | {\lambda _k}{|^p}. \end{array} $

下面估计$M_1$.当$j> k+2$时,

$ \begin{array}{l} \;\;\;\;|[b,{I_l}]{a_k}(x)|{\chi _j}(x)\\ \le |b(x) - {b_{{B_k}}}|\left| {\int_{{B_k}} {\left( {\frac{1}{{|x - y{|^{n - l}}}} - \frac{1}{{|x{|^{n - l}}}}} \right)} {a_k}(y)dy} \right|\\ \;\;\;\;\;\;\; + \left| {\int_{{B_k}} {\frac{1}{{|x - y{|^{n - l}}}}} (b(y) - {b_{{B_k}}}){a_k}(y)dy} \right|\\ \le \;C|b(x) - {b_{{B_k}}}|\int_{{B_k}} {\frac{{|y|}}{{|x{|^{n + 1 - l}}}}} |{a_k}(y)|dy + \frac{C}{{|x{|^{n - l}}}}\int_{{B_k}} | b(y) - {b_{{B_k}}}||{a_k}(y)|dy\\ \le C{2^{k - j(n + 1) + jl}}|b(x) - {b_{{B_k}}}|\int_{{B_k}} | {a_k}(y)|dy + C{2^{ - jn + jl}}\int_{{B_k}} | b(y) - {b_{{B_k}}}||{a_k}(y)|dy. \end{array} $

注意到

$ \begin{array}{l}& \int_{B_k}|a_k(y)|dy\leq C\frac{2^{kn}}{\mu(B_k)}\int_{B_k}|a_k(y)|\mu(y)dy\\ \leq &\ C2^{kn}\left(\frac{1}{\mu(B_k)}\int_{B_k}|a_k(y)|^{q_1}\mu(y)dy\right)^{1/q_1}\leq C2^{kn}\mu(B_k)^{-\frac{1}{q_{_1}}-\frac{\alpha}{n}},\end{array} $

$ \begin{array}{l} \int_{{B_k}} | b(y) - {b_{{B_k}}}||{a_k}(y)|dy\\ \le {\left( {\int_{{B_k}} | b(y) - {b_{{B_k}}}{|^{q_1^\prime }}\mu {{(y)}^{1 - q_1^\prime }}} \right)^{\frac{1}{{{{q'}_{_1}}}}}}{\left( {\int_{{B_k}} | {a_k}(y)|\mu (y)} \right)^{\frac{1}{{{q_{_1}}}}}}\\ \le C{\left\| b \right\|_{{\rm{Li}}{{\rm{p}}_\beta }(\mu )}}\mu {({B_k})^{1 + \frac{\beta }{n} - \frac{1}{{{q_{_1}}}} - \frac{\alpha }{n}}}, \end{array} $

所以

$ \begin{array}{l} |[b,{I_l}]{a_k}(x)|{\chi _j}(x)\\ \le C{2^{(k - j)(n + 1) + jl}}\mu {({B_k})^{ - \frac{1}{{{q_{_1}}}} - \frac{\alpha }{n}}}\left( {|b(x) - {b_{{B_j}}}| + |{b_{{B_j}}} - {b_{{B_k}}}|} \right)\\ \;\;\;+ C{\left\| b \right\|_{{\rm{Li}}{{\rm{p}}_\beta }}}(\mu ){2^{ - jn + jl}}\mu {({B_k})^{1 + \frac{\beta }{n} - \frac{1}{{{q_{_1}}}} - \frac{\alpha }{n}}}. \end{array} $

首先我们有

$ 2^{jlq_{_2}}\int_{B_j} |b(x)-b_{B_j}|^{q_2}\mu(x)^{1-(1-\frac{l}{n})q_{_2}}dx\leq C\|b\|^{q_2}_{{ \hbox{Lip}}_{\beta}(\mu)}\mu(B_j)^{1+\frac{l+\beta}{n}q_{_2}}. $

事实上, 取$t=(1-\frac{1}{q_2})\frac{n}{l+\beta}, $由于$\frac{1}{q_2}=\frac{1}{{{q_{_1}}}}-\frac{l+\beta}{n}, $所以$t>1$.由Hölder不等式以及$\mu\in A_1$,

$ \begin{array}{l} &2^{jlq_{_2}}\int_{B_j} |b(x)-b_{B_j}|^{q_{_2}}\mu(x)^{1-(1-\frac{l}{n})q_{_2}}dx\\ \leq \ &C2^{jlq_{_2}}\left(\int_{B_j}|b(x)-b_{B_j}|^{q_{_2}t'}\mu(x)^{1-q_{_2}t'}dx\right)^{\frac{1}{t'}}\left(\int_{B_j}\mu(x)^{1+\frac{l}{n}q_{_2}t} dx\right)^{\frac{1}{t}}\\ \leq \ &C\|b\|^{q_{_2}}_{{ \hbox{Lip}}_{\beta}(\mu)}\mu(B_j)^{\frac{1}{t'}+\frac{\beta}{n}q_{_2}}\left(\inf_{x\in B_j}\mu(x)\right)^{\frac{1}{t}+\frac{l}{n}q_{_2}}|B_j|^{\frac{1}{t}+\frac{l}{n}q_{_2}}\\ \leq\ & C\|b\|^{q_{_2}}_{{ \hbox{Lip}}_{\beta}(\mu)}\mu(B_j)^{1+\frac{l+\beta}{n}q_{_2}}.\end{array} $

再注意到

$ \int_{B_j} \mu(x)^{1-(1-\frac{l}{n})q_{_2}}dx \leq C|B_j|^{(1-\frac{l}{n})q_{_2}}\mu(B_j)^{1-(1-\frac{l}{n})q_{_2}}, $

所以, 由引理2.4, 当$j> k+2$时,

$ \begin{array}{l} & \ \mu(B_j)^{\frac{\alpha}{n}}\|[b,I_l] a_k(x)\chi_j(x)\|_{L^{q_{_2}}(\mu^{1-(1-{l}/{n})q_{_2}})}\\ \leq &\ C\|b\|_{{ \hbox{Lip}}_{_\beta}(\mu)} 2^{(k-j)(n+1)}\left(\frac{\mu(B_j)}{\mu(B_k)}\right)^{\frac{1}{q_{_1}}+\frac{\alpha}{n}}\\ &\ + C\|b\|_{{ \hbox{Lip}}_{_\beta}(\mu)} 2^{(k-j)(1-\beta)}\left(\frac{\mu(B_k)}{\mu(B_j)}\right)^{1+\frac{\beta}{n}-\frac{1}{q_{_1}}-\frac{\alpha}{n}} + C\|b\|_{{ \hbox{Lip}}_{_\beta}(\mu)} \left(\frac{\mu(B_k)}{\mu(B_j)}\right)^{1+\frac{\beta}{n}-\frac{1}{q_{_1}}-\frac{\alpha}{n}}\\ \leq &\ C\|b\|_{{ \hbox{Lip}}_{_\beta}(\mu)} \left(2^{(k-j)(n+1)}\left(\frac{\mu(B_j)}{\mu(B_k)}\right)^{\frac{1}{q_{_1}}+\frac{\alpha}{n}} +\left(\frac{\mu(B_k)}{\mu(B_j)}\right)^{1+\frac{\beta}{n}-\frac{1}{q_{_1}}-\frac{\alpha}{n}}\right)\\ \leq &\ C\|b\|_{{ \hbox{Lip}}_{_\beta}(\mu)} \left(2^{(k-j)(n(\frac{1}{q_{_1}}-1)-\alpha+1)}+2^{(k-j)\delta(n(1-\frac{1}{q_{_1}})+\beta-\alpha)}\right)\\ \leq &\ C\|b\|_{{ \hbox{Lip}}_{_\beta}(\mu)}2^{(k-j)\delta(n(1-\frac{1}{q_{_1}})+\beta-\alpha)}=C\|b\|_{{ \hbox{Lip}}_{_\beta}(\mu)}W(j,k) . \end{array} $

从而可得

$ \begin{array}{l} {M_1} \le C\left\| b \right\|_{{\rm{Li}}{{\rm{p}}_\beta }(\mu )}^p\sum\limits_{j = - \infty }^\infty {{{\left[ {\sum\limits_{k = - \infty }^{j - 3} | {\lambda _k}|W(j,k)} \right]}^p}} \\ \;\;\;\; \le C\left\| b \right\|_{{\rm{Li}}{{\rm{p}}_\beta }(\mu )}^p\left\{ {\begin{array}{*{20}{l}} {\sum\limits_{j = - \infty }^\infty {\sum\limits_{k = - \infty }^{j - 3} | } {\lambda _k}{|^p}W{{(j,k)}^p},当0 < p \le 1}&{}\\ {\sum\limits_{j = - \infty }^\infty {\left[ {\sum\limits_{k = - \infty }^{j - 3} | {\lambda _k}{|^p}W(j,k)} \right]} {{\left[ {\sum\limits_{k = - \infty }^{j - 2} W (j,k)} \right]}^{p/{p^\prime }}},当p > 1}&{} \end{array}} \right.\;\\ \;\;\;\; \le C\left\| b \right\|_{{\rm{Li}}{{\rm{p}}_\beta }(\mu )}^p\sum\limits_{k = - \infty }^\infty | {\lambda _k}{|^p}. \end{array} $

综合对$M_1$和$M_2$的估计, 得

$ {\left\| {[b,{I_l}]f} \right\|_{\mathop {K_{{q_2}}^{\alpha ,p}}\limits^. (\mu ,\mu 1 - {{(1 - l/n)}_{{q_2}}})}} \le C\left\| b \right\|_{{\rm{Li}}{{\rm{p}}_\beta }(\mu )}^p{\left( {\sum\limits_{k = - \infty }^\infty | {\lambda _k}{|^p}} \right)^{\frac{1}{p}}}. $

再对$f$的所有中心原子分解取下确界即完成证明.

设$f=\sum\limits_{k=-\infty}^{\infty}\lambda_ka_k$, $a_k$是支集为$B_k$的中心$(\alpha, q_{_1};\mu, \mu)$ -原子, $\sum\limits_{k=-\infty}^{\infty}|\lambda_k|^p<\infty, $则

$ \begin{array}{l} \left\| {[b,{I_l}]f} \right\|_{W\dot K_{{q_2}}^{\alpha ,p}(\mu ,\mu 1 - (1 - l/n){q_2})}^p\\ \le \;C{\lambda ^p}\sum\limits_{j = - \infty }^\infty \mu {({B_j})^{\frac{{\alpha p}}{n}}}{\mu ^{1 - (1 - l/n){q_2}}}{(\{ x \in {C_j}:|\sum\limits_{k = - \infty }^{j - 3} {{\lambda _k}} (b(x) - {b_{{B_k}}}){I_l}{a_k}(x)| > \frac{\lambda }{3}\} )^{p/{q_2}}}\\ \;\;\;\; + \;C{\lambda ^p}\sum\limits_{j = - \infty }^\infty \mu {({B_j})^{\frac{{\alpha p}}{n}}}{\mu ^{1 - (1 - l/n){q_2}}}{(\{ x \in {C_j}:|\sum\limits_{k = - \infty }^{j - 3} {{\lambda _k}} {I_l}((b - {b_{{B_k}}}){a_k})(x)| > \frac{\lambda }{3}\} )^{p/{q_2}}}\\ \;\;\;\; + \;C{\lambda ^p}\sum\limits_{j = - \infty }^\infty \mu {({B_j})^{\frac{{\alpha p}}{n}}}{\mu ^{1 - (1 - l/n){q_2}}}{(\{ x \in {C_j}:|\sum\limits_{k = j - 2}^\infty {{\lambda _k}} [b,{I_l}]{a_k}(x)| > \frac{\lambda }{3}\} )^{p/{q_2}}}\\ = {N_1} + {N_2} + {N_3} \end{array} $

由对$M_2$的估计,

$\begin{array}{l} N_3\leq\ C\sum\limits_{j = - \infty }^\infty \mu {({B_k})^{\frac{{\alpha p}}{n}}}\Big\|\sum\limits_{k = j - 2}^\infty {{\lambda _k}} ([b,{I_l}]{a_k}){\chi _j}\Big\|^p_{L^{q_{_2}}(\mu^{1-(1-l/n)q_{_2}})}\\ \;\;\;\;\;\leq \ C\|b\|^p_{{ \hbox{Lip}}_{_\beta}(\mu)}\sum\limits_{k = - \infty }^\infty | {\lambda _k}{|^p}. \end{array} $

注意到$\alpha=n(1-\frac{1}{{{q_{_1}}}})+\beta, $类似于$M_1$中的估计, 当$j> k+2$时,

$ \begin{array}{l} \mu(B_j)\big\|(b-b_{B_k})I_la_k\chi_j\big\|_{L^{q_{_2}}(\mu^{1-(1-l/n)q_{_2}})}\leq C\|b\|_{{ \hbox{Lip}}_{_\beta}(\mu)} 2^{(k-j)(1-\beta)}, \end{array} $

所以

$ \begin{array}{l} \ N_1\leq \ C\sum\limits_{j = - \infty }^\infty \mu ({B_k})^{\frac{\alpha p}{n}} \left(\sum\limits_{k = j - 2}^\infty | {\lambda _k}|\left\| {(b - {b_{{B_k}}}){I_l}{a_k}{\chi _j}} \right\|_{L^{q_{_2}}(\mu^{1-(1-l/n)q_{_2}})}\right)^p\\ \;\;\;\;\;\;\leq \ C\sum\limits_{j = - \infty }^\infty {\sum\limits_{k = j - 2}^\infty | } |\lambda_k|^p2^{(k-j)(1-\beta)p} \leq C\sum\limits_{k = - \infty }^\infty | |\lambda_k|^p. \end{array} $

再注意到, 当$x\in C_j, j> k+2$时,

$ \begin{array}{l} I_l((b-b_{B_k})a_k)(x)\leq\ \left|\int_{B_k}\frac{1}{|x-y|^{n-l}}(b(y)-b_{B_k})a_k(y)dy\right|\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\leq\ C2^{-j(n-l)}\|b\|_{{ \hbox{Lip}}_{_\beta}(\mu)} \mu(B_k)^{1+\frac{\beta}{n}-\frac{1}{q_{_1}}-\frac{\alpha}{n}}\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\ C2^{-j(n-l)}\|b\|_{{ \hbox{Lip}}_{_\beta}(\mu)}, \end{array} $

所以, 当$x\in C_j$时,

$ \begin{array}{l} \left|\mathop \sum \limits_{k = - \infty }^{j - 3} \lambda_kI_l(b-b_{B_k})a_k(x)\right| \leq \ C2^{-(n-l)}\|b\|_{{ \hbox{Lip}}_{_\beta}(\mu)} \mathop \sum \limits_{k = - \infty }^{j - 3} |\lambda_k|\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\leq\ C2^{-j(n-l)}\|b\|_{{ \hbox{Lip}}_{_\beta}(\mu)}\sum\limits_{k = - \infty }^\infty | {\lambda _k}|. \end{array} $

可选取$j_0\in\mathbb{Z}$, 使得

$ \begin{array}{l} 2^{j_0(n-l)}\leq 3C^{-1}\lambda \|b\|_{{ \hbox{Lip}}_{_\beta}(\mu)}\mathop \sum \limits_{k = - \infty }^\infty |\lambda_k|\leq 2^{(j_0+1)(n-l)} \end{array} $

成立.易见, 若$j\geq j_0+1, $则

$ \Big\{x\in C_j:\big|\sum\limits_{k=-\infty}^{j-3}\lambda_kI_l((b-b_{B_k})a_k)(x)\big|>\frac{\lambda}{3}\Big\}=\phi, $

注意到$\mu^{1-(1-l/n)q_{_2}}(B_j)\leq C|B_j|^{(1-l/n)q_{_2}}\mu(B_j)^{1-(1-l/n)q_{_2}}$以及$1/q_2=1/q_1-(l+\beta) /n, \alpha=n(1-1/q_1)+\beta, 0<p\leq 1$, 因此

$ \begin{array}{l} N_2\leq \ C\mathop {\sup }\limits_{\lambda > 0} \lambda^p\mathop \sum \limits_{j = - \infty }^{{j_0}} \mu(B_j)^{\frac{\alpha p}{n}}(\mu^{1-(1-l/n)q_{_2}}(B_j))^{p/q_{_2}}\\ \;\;\;\;\;\;\;\leq \ C\mathop {\sup }\limits_{\lambda > 0} \lambda^p\sum\limits_{j = - \infty }^{{j_0}} {{2^{j(n - l)p}}} \leq C\mathop {\sup }\limits_{\lambda > 0} \lambda^p 2^{j_0(n-l) p}\\ \;\;\;\;\;\;\leq\ C\|b\|^p_{{ \hbox{Lip}}_{_\beta}(\mu)}\left(\mathop \sum \limits_{k = - \infty }^\infty |\lambda_k|\right)^p\leq C\|b\|^p_{{ \hbox{Lip}}_{_\beta}(\mu)}\mathop \sum \limits_{k = - \infty }^\infty |\lambda_k|^p. \end{array} $

综合$N_1, N_2, N_3$的估计,

$ \begin{array}{l} \|[b,I_l]f\|^p_{W\dot{K}_{q_{_2}}^{\alpha,p}(\mu,\mu^{1-(1-l/n)q_{_2}})}\leq C\|b\|_{{ \hbox{Lip}}_{_\beta}(\mu)}\left(\mathop \sum \limits_{k = - \infty }^\infty | \lambda_k|^p\right)^{1/p}. \end{array} $

再对$f$的所有中心原子分解取下确界即完成定理1.2的证明.