Heisenberg群上与Schr?dinger算子有关的交换子的紧性

Compactness of Commutators Related with Schr?dinger Operators on Heisenberg Groups

摘要: 让L表示一个Schrӧdinger算子。本文研究了Heisenberg群上与L有关的交换子的紧性问题。通过光滑截断技术，证明了Heisenberg群上分数Schrӧdinger热半群生成的积分算子关于消失平均震荡型空间CMO(ρ)(Hⁿ)中函数的交换子是紧算子。作为应用，讨论了与L有关的极大函数交换子的紧性。

Abstract: Let L denote the Schrӧdinger operator. In this paper, we study the result of compactness of commutators related to L on Heisenberg groups. By smooth truncation technique, we show that commutators of the maximal operators generated by fractional Schrӧdinger heat semi groups with the function in the vanishing mean oscillation type space CMO(ρ)(Hⁿ) are compact operators. As an application, we investigate compactness of commutators of maximal functions associated with L.

1. 引言

若存在常数 $C>0$，使得一个非负局部 $L^q$ 可积的函数V满足逆Hӧlder不等式

${\left(\frac{1}{\left|B\right|}{\displaystyle {\int }_B{V}^q\left(h\right)\text{d}h}\right)}^{1/q}\le \frac{C}{\left|B\right|}{\displaystyle {\int }_{B}V\left(h\right)\text{d}h}.$

则称V属于逆Hӧlder类 $B_q,q>\left(2n+2\right)/2$。我们用符号 $\vartheta =2n+2$ 表示海森堡群的齐次维数。令 $L=-{\Delta }_{H^n}+V$ 是Heisenberg群上的一个Schrӧdinger算子，其中 ${\Delta }_{H^n}$ 是次Laplace算子，也就是 ${\Delta }_{H^n}:={\displaystyle {\sum }_{j=1}^{2n}\left(X_j^2\right)}$。且非负势 $V\in B_q,q>\vartheta /2$。积分算子T关于局部可积的函数b的交换子 $\left[b,T\right]$ 被定义为 $\left[b,T\right]f=bT\left(f\right)-T\left(bf\right)$，其中f为任意光滑函数。

在泛函分析中，一个重要的分支是紧算子理论。设T是从一个Banach空间X到另一个Banach空间Y的线性算子。如果X的任意有界子集在算子T作用下的像是Y中的紧子集，我们称T为紧算子。紧算子的一个经典例子是Sobolev空间的紧嵌入。通过这种嵌入，可以将椭圆型边值问题转化为Fredholm积分方程。有关紧算子的更多信息可以参考 [1] 和 [2]。近年来，与Schrӧdinger算子L相关的交换子紧性的研究引起了越来越多的注意。具体地，P.T. Li和L.Z. Peng在 [3] 中给出了与L相关的Riesz变换的一些交换子的紧性问题。P.T. Li，Y. M和C.Y. Zhang在 [4] 中建立了一个紧性准则，并应用于与L有关的Riesz变换的交换子。在 [5] 中，Q. He和P. Li讨论了与L有关的标准Calderόn-Zygmund算子、Riesz变换和Littlewood-Paley函数等的交换子的加权紧性问题。对于更多的紧性问题和相关结果，参考文献 [6] 和 [7]。

本文不同于 [3] [4] [5] 中的研究对象，主要研究了分数Schrӧdinger热半群生成的极大算子交换子的紧性。首先，定义了一族满足新的核估计与L有关的算子 ${\left\{{\Phi }_{t,{\gamma }_1,{\gamma }_2}^L\right\}}_{t\ge 0}$，再定义其极大算子 ${\Phi }_{{\gamma }_1,{\gamma }_2}^{L,*}$ 及其交换子 $\left[b,{\Phi }_{{\gamma }_1,{\gamma }_2}^{L,*}\right]$，其中 $b\in CMO\left(\rho \right)\left(H^n\right)$。先讨论极大算子及其交换子的 $L^p$ 有界性，进一步地，通过光滑截断技术方法和Frechet-Kolomogorv定理来得到Heisenberg群上 $\left[b,{\Phi }_{{\gamma }_1,{\gamma }_2}^{L,*}\right]$ 的 $L^p$ 紧性。光滑截断技术方法的基本思想是给出一族截断算子并借助一些算子的核估计和一些新的极大算子的 $L^p$ 有界性来证明交换子的紧性问题。值得指出的是，我们在定理4.1中得到的 $\left[b,{\Phi }_{{\gamma }_1,{\gamma }_2}^{L,*}\right]$ 的 $L^p$ 紧性涵盖了很多与L有关的极大函数。作为应用，我们选取了与分数次热半群算子核和广义的泊松算子有关的积分核 $t^m{\partial }_t^m{K}_{\alpha ,t}^L\left(\cdot ,\cdot \right)$ 和 $t^{\nu }{\partial }_t^{\nu }{p}_{t,\sigma }^{L,\alpha }\left(\cdot ,\cdot \right)$，通过分数阶热核的正则性估计可以验证这些积分核满足所定义的新核估计，从而得出这两种核所对应的极大函数对于 $b\in CMO\left(\rho \right)\left(H^n\right)$ 的交换子 $\left[b,T_i^{L,*}\right],i=1,2$ 是 $L^p$ 紧算子。下面，将介绍Heisenberg群的一些基本性质。

$2n+1$ 维Heisenberg群 $H^n$ 是具有基本流形 $R^{2n}\times R^n$ 和如下乘法的Lie群：

$\left(x,t\right)\left(y,s\right):=\left(x+y,t+s+2{\displaystyle \underset{j=1}{\overset{n}{\sum }}\left(x_{n+j}{y}_j-x_j{y}_{n+j}\right)}\right).$

$H^n$ 上的左不变向量场的Lie代数由下式给出：

$X_{2n+1}=\frac{\partial }{{\partial }_t},X_j=\frac{\partial }{{\partial }_{x_j}}+2x_{n+j}\frac{\partial }{{\partial }_t},X_{n+j}=\frac{\partial }{{\partial }_{x_{n+j}}}+2x_j\frac{\partial }{{\partial }_t},j=1,2,\cdots ,n.$

$H^n$ 上的梯度定义为 ${\nabla }_{H^n}:=\left(X_1,\cdots ,X_{2n}\right)$。其左不变距离为： $d\left(h,g\right):=\left|h^{-1}g\right|$。以g为中心，半径r的球表示为 $B\left(g,r\right)=\left\{h:\left|h^{-1}g\right|<r\right\}$。该球的体积为： $\left|B\left(g,r\right)\right|=c_n{r}^n$，其中 $c_n$ 表示 $H^n$ 上单位球的体积。

本文内容如下：第2节叙述了一些与L有关的核估计。第3节给出了与L有关的极大函数 ${\Phi }_{{\gamma }_1,{\gamma }_2}^{L,*}$ 及其交换子 $\left[b,{\Phi }_{{\gamma }_1,{\gamma }_2}^{L,*}\right]$ 的有界性的证明，这对后续的紧性证明非常重要。作为本文的主要结果，在第4节中通过光滑截断技术建立了 $H^n$ 上 $\left[b,{\Phi }_{{\gamma }_1,{\gamma }_2}^{L,*}\right]$ 的 $L^p$ 紧性，其中 $b\in CMO\left(\rho \right)\left(H^n\right)$。作为应用，进一步得出 $T_i^{L,*},i=1,2$，对于 $b\in CMO\left(\rho \right)\left(H^n\right)$ 的极大交换子是 $L^p$ 紧算子。

2. 一些核估计

Z.W. Shen在 [8] 中首次引入了辅助函数 $\rho (\cdot )$。 $H^n$ 上的辅助函数 $\rho (\cdot )$ 可以类似地如下定义。

定义2.1函数 $\rho (\cdot )$ 被定义为 $\rho \left(g\right):=\underset{r>0}{\sup }\left\{r:r^{-\left(\vartheta -2\right)}{\displaystyle {\int }_{B\left(g,r\right)}V\left(h\right)\text{d}h\le 1}\right\},g\in H^n$。

引理2.2 [9] 假设 $V\in B_q$，其中 $q>\vartheta /2$，则存在常数 $k_0>0$， $C_0>1$ 和 $C>0$ 使得

$\frac{\rho \left(g\right)}{C_0}{\left(1+\left|g^{-1}h\right|/\rho \left(g\right)\right)}^{-k_0}\le \rho \left(h\right)\le C_0\rho \left(g\right){\left(1+\left|g^{-1}h\right|/\rho \left(g\right)\right)}^{k_0/\left(k_0+1\right)}.$ (1)

特别地，当 $\left|g^{-1}h\right|\le C\rho \left(g\right)$ 时，有 $\rho \left(g\right)~\rho \left(h\right)$ 成立。

本文中，定义 ${\Psi }_{\theta }\left(B\right)={\left(1+r/\rho \left(g\right)\right)}^{\theta }$，其中 $B=B\left(g,r\right)$ 是中心为g，半径为r的一个球，且 $\theta >0$。特别地，对于给定的 $\theta >0$，用符号 $\Psi \left(B\right)$ 来代替 ${\Psi }_{\theta }\left(B\right)$。下面分别给出空间 $BMO\left(\rho \right)\left(H^n\right)$ 和 $CMO\left(\rho \right)\left(H^n\right)$ 的定义。

定义2.3 1) 对 $\theta >0$， $BMO\left(\rho \right)\left(H^n\right)$ 空间被定义为满足下列条件的局部可积函数f的集合。对所有的 $g\in H^n$ 和 $r>0$，有 $\frac{1}{\left|B\right|}{\displaystyle {\int }_B\left|f\left(h\right)-f_B\right|\text{d}h}\le C{\Psi }_{\theta }\left(B\right)$，其中 $f_B=\frac{1}{\left|B\right|}{\displaystyle {\int }_B\left|f\left(h\right)\right|\text{d}h}$。

$BMO\left(\rho \right)\left(H^n\right)$ 空间中函数的范数被定义为 ${\Vert f\Vert }_{BMO\left(\rho \right)}:=\underset{B\subset H^n}{\sup }\frac{1}{{\Psi }_{\theta }\left(B\right)\left|B\right|}{\displaystyle {\int }_B\left|f\left(g\right)-f_B\right|\text{d}g}<\infty$。

2) $CMO\left(\rho \right)\left(H^n\right)$ 表示为 $C_c^{\infty }$ 在 $BMO\left(\rho \right)\left(H^n\right)$ 的拓扑空间中的闭包，其中 $C_c^{\infty }$ 是 $H^n$ 上具有紧支撑的无穷次可微函数的集合。

特殊的极大算子 $M_{V,\eta }\left(0<p<\infty \right)$ 被定义为 $M_{V,\eta }f\left(g\right):=\underset{g\in B}{\sup }\frac{1}{{\Psi }_{\theta }{\left(B\right)}^{\eta }\left|B\right|}{\displaystyle {\int }_B\left|f\left(h\right)\right|\text{d}h}$。

因为 $M_{V,\eta }f$ 比经典的Hardy-Littlewood极大函数要小，所以下面的结论是显然成立的。

引理2.4让 $1<p<\infty$ 而且 $p^{\prime }=p/\left(p+1\right)$，则存在一个常数 $C>0$ 使得 ${\Vert M_{V,p^{\prime }}f\Vert }_{L^p\left(H^n\right)}\le C{\Vert f\Vert }_{L^p\left(H^n\right)}$。

$M_{V,\eta }$ 的交换子被定义为 $\left[b,M_{V,\eta }\right]\left(f\right)\left(g\right):=\underset{g\in B}{\sup }\frac{1}{{\Psi }_{\theta }{\left(B\right)}^{\eta }\left|B\right|}{\displaystyle {\int }_B\left|b\left(g\right)-b\left(h\right)\right|\left|f\left(h\right)\right|\text{d}h}$。

让 $p_{t,\sigma }^{L,\alpha }$ 表示与L相关广义的泊松算子，即 $p_{t,\sigma }^{L,\alpha }\left(g,h\right):=\frac{1}{\Gamma \left(\sigma \right)}{\displaystyle {\int }_0^{\infty }{\text{e}}^{-r}{K}_{\alpha ,t^{2\alpha }/\left(4r\right)}^L}\left(g,h\right)\frac{\text{d}r}{r^{1-\sigma }},0<\sigma <1$。

与L有关的分数次热半群表示为 ${\left\{{\text{e}}^{-t{L}^{\alpha }}\right\}}_{t\ge 0},\alpha \in \left(0,1\right)$。通过从属公式(参考 [10] )，其分式热核 $K_{\alpha ,t}^L\left(\cdot ,\cdot \right)$ 为 $K_{\alpha ,t}^L\left(g,h\right)={\displaystyle {\int }_0^{\infty }{\eta }_t^{\alpha }\left(s\right)K_s^L\left(g,h\right)\text{d}s}$，其中 $K_s^L\left(\cdot ,\cdot \right)$ 是热半群 ${\left\{{\text{e}}^{-sL}\right\}}_{s\ge 0}$ 的积分核， ${\eta }_t^{\alpha }(\cdot )$ 是 $\left(0,\infty \right)$ 上的连续函数，且满足

$\{\begin{array}{l}{\eta }_t^{\alpha }\left(s\right)=t^{-1/\alpha }{\eta }_1^{\alpha }\left(s/t^{1/\alpha }\right);\\ {\eta }_t^{\alpha }\left(s\right)\le Ct/\left(s^{1+\alpha }\right)\forall s,t>0;\\ {\displaystyle {\int }_0^{\infty }{s}^{-\gamma }{\eta }_t^{\alpha }\left(s\right)\text{d}s}<\infty ,\gamma >0;\\ {\eta }_t^{\alpha }\left(s\right)\simeq t/\left(s^{1+\alpha }\right)\forall s\ge t^{1/\alpha }>0.\end{array}$

定义2.5算子 $T_{i,t}^L$ 被定义为 $T_{i,t}^L\left(f\right)\left(g\right):={\displaystyle {\int }_{H^n}{K}_i^L\left(g,h\right)f\left(h\right)\text{d}h},i=1,2$，其中

$\{\begin{array}{l}{K}_1^L\left(g,h\right):=t^m{\partial }_t^m{K}_{\alpha ,t}^L\left(g,h\right),m>0;\\ K_2^L\left(g,h\right):=t^{\nu }{\partial }_t^{\nu }{p}_{t,\sigma }^{L,\alpha }\left(g,h\right),\nu >0.\end{array}$

$T_{i,t}^L$ 的极大算子 $T_i^{L,*}$ 被定义为 $T_i^{L,*}\left(f\right)\left(g\right):=\underset{t>0}{\sup }\left|T_{i,t}^L\left(f\right)\left(g\right)\right|,i=1,2$。

$T_i^{L,*}$ 关于 $b\in BMO\left(\rho \right)\left(H^n\right)$ 的交换子被定义为

$\left[b,T_i^{L,*}\right]\left(f\right)\left(g\right):=\underset{t>0}{\sup }\left|{\displaystyle {\int }_{H^n}{K}_i^L\left(g,h\right)\left(b\left(g\right)-b\left(h\right)\right)f\left(h\right)\text{d}h}\right|,i=1,2.$

定义2.6假设 ${\gamma }_1>0$ 且 ${\gamma }_2\in \left(0,2\right)$。让 ${\left\{{\Phi }_{t,{\gamma }_1,{\gamma }_2}^L\right\}}_{t\ge 0}$ 是与L相关的积分核为 $K_{{\gamma }_2,t}^{L,{\gamma }_1}\left(\cdot ,\cdot \right)$ 的一族算子。即：

${\Phi }_{t,{\gamma }_1,{\gamma }_2}^L\left(f\right)\left(g\right):={\displaystyle {\int }_{{{\displaystyle H}}^n}{K}_{{\gamma }_2,t}^{L,{\gamma }_1}\left(g,h\right)f\left(h\right)\text{d}h}$，其中 $K_{{\gamma }_2,t}^{L,{\gamma }_1}\left(\cdot ,\cdot \right)$ 满足下面两个条件：

1) 对任意的 $N>0$，存在一个常数 $C_N>0$ 使得

$\left|K_{{\gamma }_2,t}^{L,{\gamma }_1}\left(g,h\right)\right|\le \frac{C_N{t}^{{\gamma }_1}}{{\left(t^{1/{\gamma }_2}+\left|g^{-1}h\right|\right)}^{\vartheta +{\gamma }_1{\gamma }_2}}{\left(1+\frac{t^{1/{\gamma }_2}}{\rho \left(g\right)}+\frac{t^{1/{\gamma }_2}}{\rho \left(h\right)}\right)}^{-N}.$ (2)

2) 对任意的 $N>0$，存在一个常数 $C_N>0$ 使得对每个 $0<\delta <\min \left\{{\gamma }_2,1-Q/q\right\}$ 和所有 $\left|u\right|\le t^{1/{\gamma }_2}$ 有

$\left|K_{{\gamma }_2,t}^{L,{\gamma }_1}\left(gu,h\right)-K_{{\gamma }_2,t}^{L,{\gamma }_1}\left(g,h\right)\right|\le \frac{C_N{t}^{{\gamma }_1}}{{\left(t^{1/{\gamma }_2}+\left|g^{-1}h\right|\right)}^{\vartheta +{\gamma }_1{\gamma }_2}}{\left(\frac{\left|u\right|}{t^{1/{\gamma }_2}}\right)}^{\delta }{\left(1+\frac{t^{1/{\gamma }_2}}{\rho \left(g\right)}+\frac{t^{1/{\gamma }_2}}{\rho \left(h\right)}\right)}^{-N}.$ (3)

${\Phi }_{t,{\gamma }_1,{\gamma }_2}^L$ 的极大算子 ${\Phi }_{{\gamma }_1,{\gamma }_2}^{L,*}$ 被定义为 ${\Phi }_{{\gamma }_1,{\gamma }_2}^{L,*}\left(f\right)\left(g\right):=\underset{t>0}{\sup }\left|{\Phi }_{t,{\gamma }_1,{\gamma }_2}^L\left(f\right)\left(g\right)\right|$。

${\Phi }_{{\gamma }_1,{\gamma }_2}^{L,*}$ 关于 $b\in BMO\left(\rho \right)\left(H^n\right)$ 的交换子被定义为

$\left[b,{\Phi }_{{\gamma }_1,{\gamma }_2}^{L,*}\right]\left(f\right)\left(g\right):=\underset{t>0}{\sup }\left|{\displaystyle {\int }_{H^n}{K}_{{\gamma }_2,t}^{L,{\gamma }_1}\left(g,h\right)\left(b\left(g\right)-b\left(h\right)\right)f\left(h\right)\text{d}h}\right|.$

引理2.7 [11] 让 $\alpha \in \left(0,1\right)$， $m>0$ 且 $V\in B_q,q>\vartheta /2$。

1) 对任意的 $N>0$，存在一个常数 $C_N>0$ 使得

$\left|t^m{\partial }_t^m{K}_{\alpha ,t}^L\left(g,h\right)\right|\le \frac{C_N{t}^m}{{\left(t^{1/\left(2\alpha \right)}+\left|g^{-1}h\right|\right)}^{\vartheta +2\alpha m}}{\left(1+\frac{t^{1/\left(2\alpha \right)}}{\rho \left(g\right)}+\frac{t^{1/\left(2\alpha \right)}}{\rho \left(h\right)}\right)}^{-N}.$

2) 让 $0<\delta \le \min \left\{2\alpha ,{\delta }_0\right\}$。对每个 $N>0$，这存在一个常数 $C_N>0$ 使得对所有的 $\left|u\right|\le t^{1/\left(2\alpha \right)}$，

$\left|t^m{\partial }_t^m{K}_{\alpha ,t}^L\left(gu,h\right)-t^m{\partial }_t^m{K}_{\alpha ,t}^L\left(g,h\right)\right|\le \frac{C_N{t}^m}{{\left(t^{1/\left(2\alpha \right)}+\left|g^{-1}h\right|\right)}^{\vartheta +2\alpha m}}{\left(\frac{\left|u\right|}{t^{1/\left(2\alpha \right)}}\right)}^{\delta }{\left(1+\frac{t^{1/\left(2\alpha \right)}}{\rho \left(g\right)}+\frac{t^{1/\left(2\alpha \right)}}{\rho \left(h\right)}\right)}^{-N}.$

引理2.8 [12] 让一列紧算子序列 $\left\{T_n\right\}$ 一致拓扑收敛到一个算子T，即 $\underset{n\to \infty }{\lim }\Vert T_n-T\Vert =0$。则T也是紧算子。

引理2.9 [12] 假设 $0<p<\infty$ 且 $\mathbb{F}$ 是 $L^p\left(H^n\right)$ 中的一个子集。若下面(1)~(3)成立，则 $\mathbb{F}$ 在 $L^p\left(H^n\right)$ 中是列紧的。

1) $\mathbb{F}$ 是有界的，即 $\underset{f\in \mathbb{F}}{\sup }{\Vert f\Vert }_{L^p\left(H^n\right)}<\infty$ ；

2) $\mathbb{F}$ 在无穷远处一致收敛到0，即 $\underset{N\to \infty }{lim}\underset{f\in \mathbb{F}}{\sup }{\displaystyle {\int }_{\left|g\right|>N}{\left|f\left(g\right)\right|}^p\text{d}g}=0$ ；

3) $\mathbb{F}$ 是一致等连续的，即 $\underset{\left|u\right|\to \infty }{lim}\underset{f\in \mathbb{F}}{\sup }{\displaystyle {\int }_{H^n}{\left|f\left(gu\right)-f\left(g\right)\right|}^p\text{d}g}=0$。

3. 与L相关的极大函数及其交换子的有界性

定理3.1令 $1<p<\infty$ 且 ${\gamma }_1{\gamma }_2>\theta \eta$。则存在一个常数C使得 ${\Vert {\Phi }_{{\gamma }_1,{\gamma }_2}^{L,*}\left(f\right)\Vert }_{L^p\left(H^n\right)}\le C{\Vert f\Vert }_{L^p\left(H^n\right)}$。

证明首先，把 ${\Phi }_{{\gamma }_1,{\gamma }_2}^{L,*}\left(f\right)$ 划分成两部分： ${\Phi }_{{\gamma }_1,{\gamma }_2}^{L,*}\left(f\right)\left(g\right)\le \underset{t>0}{\sup }\left\{{\displaystyle {\int }_{H^n}\left|K_{{\gamma }_2,t}^{L,{\gamma }_1}\left(g,h\right)\right|\left|f\left(h\right)\right|\text{d}h}\right\}\le I\left(g\right)+II\left(g\right)$，其中

$\{\begin{array}{l}I\left(g\right):=\underset{t>0}{\sup }\left\{{\displaystyle {\int }_{\left|g^{-1}h\right|<\rho \left(g\right)}\left|K_{{\gamma }_2,t}^{L,{\gamma }_1}\left(g,h\right)\right|\left|f\left(h\right)\right|\text{d}h}\right\};\\ II\left(g\right):=\underset{t>0}{\sup }\left\{{\displaystyle {\int }_{\left|g^{-1}h\right|\ge \rho \left(g\right)}\left|K_{{\gamma }_2,t}^{L,{\gamma }_1}\left(g,h\right)\right|\left|f\left(h\right)\right|\text{d}h}\right\}.\end{array}$

对于 $I\left(g\right)$，有 $I\left(g\right)\le I_1\left(g\right)+I_2\left(g\right)+I_3\left(g\right)$，其中

$\{\begin{array}{l}{I}_1\left(g\right):=\underset{\rho \left(g\right)>t^{1/{\gamma }_2}>0}{\sup }\left\{{\displaystyle {\int }_{\left|g^{-1}h\right|<t^{1/{\gamma }_2}}\left|K_{{\gamma }_2,t}^{L,{\gamma }_1}\left(g,h\right)\right|\left|f\left(h\right)\right|\text{d}h}\right\};\\ I_2\left(g\right):=\underset{\rho \left(g\right)>t^{1/{\gamma }_2}>0}{\sup }\left\{{\displaystyle {\int }_{t^{1/{\gamma }_2}\le \left|g^{-1}h\right|<\rho \left(g\right)}\left|K_{{\gamma }_2,t}^{L,{\gamma }_1}\left(g,h\right)\right|\left|f\left(h\right)\right|\text{d}h}\right\};\\ I_3\left(g\right):=\underset{t^{1/{\gamma }_2}\ge \rho \left(g\right)}{\sup }\left\{{\displaystyle {\int }_{\left|g^{-1}h\right|<\rho \left(g\right)}\left|K_{{\gamma }_2,t}^{L,{\gamma }_1}\left(g,h\right)\right|\left|f\left(h\right)\right|\text{d}h}\right\}.\end{array}$

通过(2)，对于 $I_1\left(g\right)$，有 $I_1\left(g\right)\le C\underset{\rho \left(g\right)>t^{1/{\gamma }_2}>0}{\sup }\left\{{\displaystyle {\int }_{\left|g^{-1}h\right|<t^{1/{\gamma }_2}}\frac{t^{{\gamma }_1}}{{\left(t^{1/{\gamma }_2}\right)}^{\vartheta +{\gamma }_1{\gamma }_2}}\left|f\left(h\right)\right|\text{d}h}\right\}\le C{M}_{V,\eta }\left(f\right)\left(g\right)$。

对于 $I_2\left(g\right)$，注意到 ${\gamma }_1{\gamma }_2>\theta \eta$，则有

$I_2\left(g\right)\le C\underset{\rho \left(g\right)>t^{1/{\gamma }_2}>0}{\sup }\left\{{\displaystyle \underset{j=1}{\overset{\infty }{\sum }}\frac{t^{{\gamma }_1}}{{\left(2^{{}^j}{t}^{1/{\gamma }_2}\right)}^{\vartheta +{\gamma }_1{\gamma }_2}}}{\displaystyle {\int }_{\left|g^{-1}h\right|\sim 2^j{t}^{1/{\gamma }_2}}\left|f\left(h\right)\right|\text{d}h}\right\}\le C{M}_{V,\eta }\left(f\right)\left(g\right).$

对于 $I_3\left(g\right)$，则有

$I_3\left(g\right)\le \underset{\rho \left(g\right)\le t^{1/{\gamma }_2}}{\sup }\left\{{\displaystyle {\int }_{\left|g^{-1}h\right|<\rho \left(g\right)}\frac{C{t}^{{\gamma }_1}\left|f\left(h\right)\right|}{{\left(t^{1/{\gamma }_2}\right)}^{\vartheta +{\gamma }_1{\gamma }_2}}\text{d}h}\right\}\le \underset{\rho \left(g\right)\le t^{1/{\gamma }_2}}{\sup }\left\{\frac{C}{{\left(t^{1/{\gamma }_2}\right)}^{\vartheta }}{\displaystyle {\int }_{\left|g^{-1}h\right|<\rho \left(g\right)}\left|f\left(h\right)\right|\text{d}h}\right\}\le C{M}_{V,\eta }\left(f\right)\left(g\right).$

对于 $II\left(g\right)$，再次通过(2)，取足够大的N，则有

$\begin{array}{c}II\left(g\right)\le C\underset{t>0}{\sup }\left\{{\displaystyle {\int }_{\left|g^{-1}h\right|\ge \rho \left(g\right)}\frac{t^{{\gamma }_1}}{{\left|g^{-1}h\right|}^{\vartheta +{\gamma }_1{\gamma }_2}}{\left(1+\frac{t^{{\gamma }_2}}{\rho \left(g\right)}\right)}^{-N}\left|f\left(h\right)\right|\text{d}h}\right\}\\ \le C\underset{t>0}{\sup }\left\{{\displaystyle \underset{j=1}{\overset{\infty }{\sum }}\frac{{\left(t^{{\gamma }_2}\right)}^{{\gamma }_1{\gamma }_2}{\left(1+t^{{\gamma }_2}/\rho \left(g\right)\right)}^{-N}}{{\left(2^j\right)}^{{\gamma }_1{\gamma }_2}{\left(\rho \left(g\right)\right)}^{{\gamma }_1{\gamma }_2}{\left(2^j\rho \left(g\right)\right)}^{\vartheta }}}{\displaystyle {\int }_{\left|g^{-1}h\right|\sim 2^j\rho \left(g\right)}\left|f\left(h\right)\right|\text{d}h}\right\}\\ \le C{M}_{V,\eta }\left(f\right)\left(g\right).\end{array}$

由引理2.4，其中 $p^{\prime }\le \eta <\infty$， $I\left(g\right)$ 和 $II\left(g\right)$ 的估计表明了

${\Vert {\Phi }_{{\gamma }_1,{\gamma }_2}^{L,*}\left(f\right)\Vert }_{L^p\left(H^n\right)}\le C{\Vert M_{V,\eta }\left(f\right)\Vert }_{L^p\left(H^n\right)}\le C{\Vert f\Vert }_{L^p\left(H^n\right)}.$

定理3.2令 $1<p<\infty$ 且 ${\gamma }_1{\gamma }_2>\theta \eta$。对 $b\in CMO\left(\rho \right)\left(H^n\right)$，存在一个常数C使得

${\Vert \left[b,{\Phi }_{{\gamma }_1,{\gamma }_2}^{L,*}\right]\left(f\right)\Vert }_{L^p\left(H^n\right)}\le C{\Vert b\Vert }_{BMO\left(\rho \right)}{\Vert f\Vert }_{L^p\left(H^n\right)}.$

证明重复定理3.1的证明过程，能得到 $\left|\left[b,{\Phi }_{{\gamma }_1,{\gamma }_2}^{L,*}\right]\left(f\right)\right|\le C\left[b,M_{V,\eta }\right]\left(f\right)\left(g\right)$。然后通过类似于 [13] 中定理1.1的证明，得出定理3.2成立。

4. 热半群极大函数的交换子的紧性

定理4.1让 $1<p<\infty$， ${\gamma }_1{\gamma }_2>\left(1+k_0\right)\theta \eta$ 且 $b\in CMO\left(\rho \right)\left(H^n\right)$，则交换子 $\left[b,{\Phi }_{{\gamma }_1,{\gamma }_2}^{L,*}\right]$ 是 $L^p\left(H^n\right)$ 上的紧算子。

证明我们将用光滑截断技术来证明该定理。取函数 $\varphi \in C^{\infty }\left(\left[0,\infty \right]\right)$ 且满足 $0\le \varphi \le 1$ 且对于 $0\le g\le 1$ 有 $\varphi \left(g\right)=1$ ；对于 $g\ge 2$ 有 $\varphi \left(g\right)=0$。对任意的 $\gamma >0$，定义 $K_{{\gamma }_2,t,\gamma }^{L,{\gamma }_1}\left(g,h\right):=K_{{\gamma }_2,t}^{L,{\gamma }_1}\left(g,h\right)\left(1-\varphi \left({\gamma }^{-1}\left|g^{-1}h\right|\right)\right)$。令

$\{\begin{array}{l}{\Phi }_{{\gamma }_1,{\gamma }_2,\gamma }^{L,*}\left(g,h\right):=\underset{t>0}{\sup }\left\{\left|{\displaystyle {\int }_{H^n}{K}_{{\gamma }_2,t,\gamma }^{L,{\gamma }_1}\left(g,h\right)f\left(h\right)\text{d}h}\right|\right\};\\ \left[b,{\Phi }_{{\gamma }_1,{\gamma }_2,\gamma }^{L,*}\right]\left(g,h\right):=\underset{t>0}{\sup }\left\{\left|{\displaystyle {\int }_{H^n}{K}_{{\gamma }_2,t,\gamma }^{L,{\gamma }_1}\left(g,h\right)}\left(b\left(g\right)-b\left(h\right)\right)f\left(h\right)\text{d}h\right|\right\}.\end{array}$ (4)

对所有的 $b\in C_c^{\infty }\left(H^n\right)$， $N=\theta \eta$ 和 $\gamma ,\eta ,\theta >0$，结合 和 ，则有

$\left|\left[b,{\Phi }_{{\gamma }_1,{\gamma }_2}^{L,*}\right]f\left(g\right)-\left[b,{\Phi }_{{\gamma }_1,{\gamma }_2,\gamma }^{L,*}\right]f\left(g\right)\right|\le C_N\left\{L_1\left(g\right)+L_2\left(g\right)\right\},$

其中

$\{\begin{array}{l}{L}_1\left(g\right):=\underset{t>0}{\sup }\left\{{\displaystyle {\int }_{\left|g^{-1}h\right|<2\gamma }\frac{t^{{\gamma }_1}}{{\left(t^{1/{\gamma }_2}+\left|g^{-1}h\right|\right)}^{\vartheta +{\gamma }_1{\gamma }_2}}{\left(1+\frac{t^{1/{\gamma }_2}}{\rho \left(g\right)}\right)}^{-N}\left|g^{-1}h\right|\left|f\left(h\right)\right|\left|\varphi \left({\gamma }^{-1}\left|g^{-1}h\right|\right)\right|\text{d}h}\right\};\\ L_2\left(g\right):=\underset{t>0}{\sup }\left\{{\displaystyle {\int }_{\left|g^{-1}h\right|\ge 2\gamma }\frac{t^{{\gamma }_1}}{{\left(t^{1/{\gamma }_2}+\left|g^{-1}h\right|\right)}^{\vartheta +{\gamma }_1{\gamma }_2}}{\left(1+\frac{t^{1/{\gamma }_2}}{\rho \left(g\right)}\right)}^{-N}\left|g^{-1}h\right|\left|f\left(h\right)\right|\left|\varphi \left({\gamma }^{-1}\left|g^{-1}h\right|\right)\right|\text{d}h}\right\}.\end{array}$

对 $L_2\left(g\right)$ 来说，因为 $\left|g^{-1}h\right|\ge 2\gamma$，所以 $\varphi \left({\gamma }^{-1}\left|g^{-1}h\right|\right)=0$。故而有

$\left|\left[b,{\Phi }_{{\gamma }_1,{\gamma }_2}^{L,*}\right]f\left(g\right)-\left[b,{\Phi }_{{\gamma }_1,{\gamma }_2,\gamma }^{L,*}\right]f\left(g\right)\right|\le C\gamma \left\{J_1\left(g\right)+J_2\left(g\right)+J_3\left(g\right)\right\},$

其中

$\{\begin{array}{l}{J}_1\left(g\right):=\underset{t^{1/{\gamma }_2}<2\gamma }{\sup }\left\{{\displaystyle {\int }_{\left|g^{-1}h\right|<t^{1/{\gamma }_2}}\frac{t^{{\gamma }_1}}{{\left(t^{1/{\gamma }_2}+\left|g^{-1}h\right|\right)}^{\vartheta +{\gamma }_1{\gamma }_2}}{\left(1+\frac{t^{1/{\gamma }_2}}{\rho \left(g\right)}\right)}^{-N}\left|f\left(h\right)\right|\text{d}h}\right\};\\ J_2\left(g\right):=\underset{t^{1/{\gamma }_2}<2\gamma }{\sup }\left\{{\displaystyle {\int }_{t^{1/{\gamma }_2}\le \left|g^{-1}h\right|<2\gamma }\frac{t^{{\gamma }_1}}{{\left(t^{1/{\gamma }_2}+\left|g^{-1}h\right|\right)}^{\vartheta +{\gamma }_1{\gamma }_2}}{\left(1+\frac{t^{1/{\gamma }_2}}{\rho \left(g\right)}\right)}^{-N}\left|f\left(h\right)\right|\text{d}h}\right\};\\ J_3\left(g\right):=\underset{t^{1/{\gamma }_2}\ge 2\gamma }{\sup }\left\{{\displaystyle {\int }_{\left|g^{-1}h\right|<2\gamma }\frac{t^{{\gamma }_1}}{{\left(t^{1/{\gamma }_2}+\left|g^{-1}h\right|\right)}^{\vartheta +{\gamma }_1{\gamma }_2}}{\left(1+\frac{t^{1/{\gamma }_2}}{\rho \left(g\right)}\right)}^{-N}\left|f\left(h\right)\right|\text{d}h}\right\}.\end{array}$

对于 $J_1\left(g\right)$，有 $J_1\left(g\right)\le C\underset{t^{1/{\gamma }_2}<2\gamma }{\sup }\left\{\frac{1}{{\left(t^{1/{\gamma }_2}\right)}^{\vartheta }}{\left(1+\frac{t^{1/{\gamma }_2}}{\rho \left(g\right)}\right)}^{-\theta \eta }{\displaystyle {\int }_{\left|g^{-1}h\right|<t^{1/{\gamma }_2}}\left|f\left(h\right)\right|\text{d}h}\right\}\le C{M}_{V,\eta }f\left(g\right)$。

对于 $J_2\left(g\right)$，有 $J_2\left(g\right)\le C\underset{t^{1/{\gamma }_2}<2\gamma }{\sup }\left\{{\displaystyle \underset{j=1}{\overset{\infty }{\sum }}\frac{t^{{\gamma }_1}}{{\left(2^j{t}^{1/{\gamma }_2}\right)}^{\vartheta +{\gamma }_1{\gamma }_2}}{\left(1+\frac{t^{1/{\gamma }_2}}{\rho \left(g\right)}\right)}^{-\theta \eta }}{\displaystyle {\int }_{\left|g^{-1}h\right|\sim 2^j{t}^{1/{\gamma }_2}}\left|f\left(h\right)\right|\text{d}h}\right\}\le C{M}_{V,\eta }f\left(g\right)$。

对于 $J_3\left(g\right)$，有 $J_3\left(g\right)\le C\underset{t^{1/{\gamma }_2}\ge 2\gamma }{\sup }\left\{\frac{1}{{\left(t^{1/{\gamma }_2}\right)}^{\vartheta }}{\left(1+\frac{2\gamma }{\rho \left(g\right)}\right)}^{-\theta \eta }{\displaystyle {\int }_{\left|g^{-1}h\right|<2\gamma }\left|f\left(h\right)\right|\text{d}h}\right\}\le C{M}_{V,\eta }f\left(g\right)$。

通过对 $J_1\left(g\right)$， $J_2\left(g\right)$ 和 $J_3\left(g\right)$ 的估计，有 $\left|\left[b,{\Phi }_{{\gamma }_1,{\gamma }_2}^{L,*}\right]f\left(g\right)-\left[b,{\Phi }_{{\gamma }_1,{\gamma }_2,\gamma }^{L,*}\right]f\left(g\right)\right|\le C{M}_{V,\eta }\left(f\right)\left(g\right)$ 成立。

由引理2.4，其中 $p^{\prime }\le \eta <\infty$，我们能得到

${\Vert \left[b,{\Phi }_{{\gamma }_1,{\gamma }_2}^{L,*}\right]\left(f\right)-\left[b,{\Phi }_{{\gamma }_1,{\gamma }_2,\gamma }^{L,*}\right]\left(f\right)\Vert }_{L^p\left(H^n\right)}\le C\gamma {\Vert M_{V,\eta }\left(f\right)\Vert }_{L^p\left(H^n\right)}\le C\gamma {\Vert f\Vert }_{L^p\left(H^n\right)},$

这表明

$\underset{\gamma \to 0}{\lim }{\Vert \left[b,{\Phi }_{{\gamma }_1,{\gamma }_2}^{L,*}\right]\left(f\right)-\left[b,{\Phi }_{{\gamma }_1,{\gamma }_2,\gamma }^{L,*}\right]\left(f\right)\Vert }_{L^p\left(H^n\right)}=0.$ (5)

另一方面，若 $b\in CMO\left(\rho \right)\left(H^n\right)$，则对任意的 $\epsilon >0$，存在 $b_{\epsilon }\in C_c^{\infty }\left(H^n\right)$ 使得 ${\Vert b-b_{\epsilon }\Vert }_{BMO\left(\rho \right)}<\epsilon$。通过定理3.2，有 ${\Vert \left[b,{\Phi }_{{\gamma }_1,{\gamma }_2}^{L,*}\right]\left(f\right)-\left[b_{\epsilon },{\Phi }_{{\gamma }_1,{\gamma }_2}^{L,*}\right]\left(f\right)\Vert }_{L^p\left(H^n\right)}\le C{\Vert b-b_{\epsilon }\Vert }_{BMO\left(\rho \right)}{\Vert {\Phi }_{{\gamma }_1,{\gamma }_2}^{L,*}\left(f\right)\Vert }_{L^p\left(H^n\right)}\le C\epsilon$。因此，为了证明

对任意的 $b\in CMO\left(\rho \right)\left(H^n\right)$，有 $\left[b,{\Phi }_{{\gamma }_1,{\gamma }_2}^{L,*}\right]$ 是 $L^p\left(H^n\right)$ 上的紧算子，只需证对所有的 $b\in C_c^{\infty }\left(H^n\right)$，有 $\left[b,{\Phi }_{{\gamma }_1,{\gamma }_2}^{L,*}\right]$ 是 $L^p\left(H^n\right)$ 上的紧算子。通过(5)和引理2.9，只需证当 $\gamma >0$ 充分小时，对所有的 $b\in C_c^{\infty }\left(H^n\right)$，有 $\left[b,{\Phi }_{{\gamma }_1,{\gamma }_2}^{L,*}\right]$ 是紧的。为了这个目的，对 $L^p\left(H^n\right)$ 中的任意有界集 $F$，取 $\mathbb{F}=\left\{\left[b,{\Phi }_{{\gamma }_1,{\gamma }_2}^{L,*}\right]\left(f\right):f\in F\right\}$。接下来，只需证对 $b\in C_c^{\infty }\left(H^n\right)$， $\mathbb{F}$ 满足引理2.10的(1)~(3)。

先证 $\mathbb{F}$ 满足(1)。根据 $K_{{\gamma }_2,t,\gamma }^{L,{\gamma }_1}$ 的定义，有 $0<\left|K_{{\gamma }_2,t,\gamma }^{L,{\gamma }_1}\left(g,h\right)\right|\le \left|K_{{\gamma }_2,t}^{L,{\gamma }_1}\left(g,h\right)\right|$。因此，

${\Phi }_{{\gamma }_1,{\gamma }_2,\gamma }^{L,*}\left(f\right)\left(g\right)\le C\underset{t>0}{\sup }\left\{{\displaystyle \underset{j=j_0}{\overset{\infty }{\sum }}{\displaystyle {\int }_{\left|g^{-1}h\right|\sim 2^j{t}^{1/{\gamma }_2}}\frac{t^{{\gamma }_1}}{{\left|g^{-1}h\right|}^{\vartheta +{\gamma }_1{\gamma }_2}}{\left(1+\frac{t^{1/{\gamma }_2}}{\rho \left(g\right)}\right)}^{-N}\left|f\left(h\right)\right|\text{d}h}}\right\}\le C{M}_{V,\eta }\left(f\right)\left(g\right),$ (6)

其中整数 $j_0$ 满足 $2^{j_0}{t}^{1/{\gamma }_2}\le \gamma \le 2^{j_0+1}{t}^{1/{\gamma }_2}$。通过这种方式，可以得到 $\left[b,{\Phi }_{{\gamma }_1,{\gamma }_2,\gamma }^{L,*}\right]\left(f\right)\left(g\right)\le C\left[b,M_{V,\eta }\right]\left(f\right)\left(g\right)$。

从定理3.1和3.2，可以推出 ${\Phi }_{{\gamma }_1,{\gamma }_2,\gamma }^{L,*}$ 和 $\left[b,{\Phi }_{{\gamma }_1,{\gamma }_2,\gamma }^{L,*}\right]$ 在 $L^p\left(H^n\right)$ 上是有界的。故有 $\underset{f\in F}{\sup }{\Vert \left[b,{\Phi }_{{\gamma }_1,{\gamma }_2,\gamma }^{L,*}\right]\left(f\right)\Vert }_{L^p\left(H^n\right)}\le C\underset{f\in F}{\sup }{\Vert f\Vert }_{L^p\left(H^n\right)}\le C_F$。这表明集合 $\mathbb{F}$ 是有界的。

再证 $\mathbb{F}$ 满足(2)。假设 $b\in C_c^{\infty }\left(H^n\right)$ 且b的支集属于 $B\left(0,R\right)$，其中 $B\left(0,R\right)$ 是一个半径为R，中心为原点的一个球。对任意 $\left|g\right|>A>2R$， $1<p<\infty$ 和 $f\in F$，应用(2)，能得到

$\left|\left[b,{\Phi }_{{\gamma }_1,{\gamma }_2,\gamma }^{L,*}\right]\left(f\right)\left(g\right)\right|\le C{\Vert b\Vert }_{L^{\infty }\left(H^n\right)}\underset{t>0}{\sup }\left\{{\displaystyle {\int }_{\left|h\right|<R}\frac{t^{{\gamma }_1}\left|f\left(h\right)\right|\text{d}h}{{\left(t^{1/{\gamma }_2}\right)}^{{\gamma }_1{\gamma }_2}{\left|g^{-1}h\right|}^{\vartheta }}}\right\}\le C{\left|g\right|}^{-\vartheta }{\displaystyle {\int }_{\left|h\right|<R}\left|f\left(h\right)\right|\text{d}h}\le C/{\left|g\right|}^{\vartheta },$

其中 $1/p^{\prime }+1/p=1$。因此， ${\displaystyle {\int }_{\left|g\right|>A}{\left|\left[b,{\Phi }_{{\gamma }_1,{\gamma }_2,\gamma }^{L,*}\right]\left(f\right)\left(g\right)\right|}^p\text{d}g}\le C{\displaystyle \underset{j=0}{\overset{\infty }{\sum }}{\displaystyle {\int }_{2^{j}A<\left|g\right|\le 2^{j+1}A}{\left(2^{j}A\right)}^{-\vartheta p}\text{d}g}}\le C/{\left|A\right|}^{\vartheta \left(p-1\right)}$。从而有 $\underset{A\to \infty }{lim}{\displaystyle {\int }_{\left|g\right|>A}{\left|\left[b,{\Phi }_{{\gamma }_1,{\gamma }_2,\gamma }^{L,*}\right]\left(f\right)\left(g\right)\right|}^p\text{d}g}=0$。

最后证 $\mathbb{F}$ 满足(3)。仅需验证对任意的 $\epsilon >0$，若 $\left|u\right|$ 是充分小的且仅取决于 $\epsilon$，对任意的 $f\in F$ 有

$\underset{\left|u\right|\to 0}{lim}{\Vert \left[b,{\Phi }_{{\gamma }_1,{\gamma }_2,\gamma }^{L,*}\right]\left(f\right)\left(\cdot u\right)-\left[b,{\Phi }_{{\gamma }_1,{\gamma }_2,\gamma }^{L,*}\right]\left(f\right)(\cdot )\Vert }_{L^p\left(H^n\right)}=C\epsilon$ (7)

成立。下面我们取 $\gamma \in \left(0,1\right)$ 和 $\left|u\right|\le \gamma /4$，则有 $\left|\left[b,{\Phi }_{{\gamma }_1,{\gamma }_2,\gamma }^{L,*}\right]\left(f\right)\left(gu\right)-\left[b,{\Phi }_{{\gamma }_1,{\gamma }_2,\gamma }^{L,*}\right]\left(f\right)\left(g\right)\right|\le I\left(g\right)+II\left(g\right),$

其中 $\{\begin{array}{l}I\left(g\right):=\underset{t>0}{\sup }\left\{{\displaystyle {\int }_{H^n}\left|K_{{\gamma }_2,t,\gamma }^{L,{\gamma }_1}\left(gu,h\right)-K_{{\gamma }_2,t,\gamma }^{L,{\gamma }_1}\left(g,h\right)\right|\left|b\left(gu\right)-b\left(h\right)\right|\left|f\left(h\right)\right|\text{d}h}\right\};\\ II\left(g\right):=\underset{t>0}{\sup }\left\{{\displaystyle {\int }_{H^n}\left|K_{{\gamma }_2,t,\gamma }^{L,{\gamma }_1}\left(g,h\right)\right|\left|b\left(gu\right)-b\left(g\right)\right|\left|f\left(h\right)\right|\text{d}h}\right\}.\end{array}$

事实上，

$\begin{array}{l}\left|K_{{\gamma }_2,t,\gamma }^{L,{\gamma }_1}\left(gu,h\right)\left(b\left(gu\right)-b\left(h\right)\right)f\left(h\right)-K_{{\gamma }_2,t,\gamma }^{L,{\gamma }_1}\left(g,h\right)\left(b\left(g\right)-b\left(h\right)\right)f\left(h\right)\right|\\ \le \left|\left(K_{{\gamma }_2,t,\gamma }^{L,{\gamma }_1}\left(gu,h\right)-K_{{\gamma }_2,t,\gamma }^{L,{\gamma }_1}\left(g,h\right)\right)\left(b\left(gu\right)-b\left(h\right)\right)f\left(h\right)\right|+\left|K_{{\gamma }_2,t,\gamma }^{L,{\gamma }_1}\left(g,h\right)\left(b\left(gu\right)-b\left(g\right)\right)f\left(h\right)\right|.\end{array}$

先考虑 $II\left(g\right)$，通过(6)，能得到 $II\left(g\right)\le {\Vert b\Vert }_{L^{\infty }\left(H^n\right)}\left|u\right|\underset{t>0}{\sup }\left\{{\displaystyle {\int }_{H^n}\left|K_{{\gamma }_2,t,\gamma }^{L,{\gamma }_1}\left(g,h\right)\right|\left|f\left(h\right)\right|\text{d}h}\right\}\le C\left|u\right|M_{V,\eta }\left(f\right)\left(g\right)$。

通过引理2.4，其中 $p^{\prime }\le \eta <\infty$，有 ${\Vert II\Vert }_{L^p\left(H^n\right)}\le C\left|u\right|{\Vert M_{V,\eta }\left(f\right)\Vert }_{L^p\left(H^n\right)}\le C\left|u\right|{\Vert f\Vert }_{L^p\left(H^n\right)}$。

接下来，对 $I\left(g\right)$，有 $I\left(g\right)\le I_1\left(g\right)+I_2\left(g\right)$，其中

$\{\begin{array}{l}{I}_1\left(g\right):=\underset{t^{1/{\gamma }_2}\ge \left|u\right|}{\sup }\left\{{\displaystyle {\int }_{H^n}\left|K_{{\gamma }_2,t,\gamma }^{L,{\gamma }_1}\left(gu,h\right)-K_{{\gamma }_2,t,\gamma }^{L,{\gamma }_1}\left(g,h\right)\right|\left|b\left(gu\right)-b\left(h\right)\right|\left|f\left(h\right)\right|\text{d}h}\right\};\\ I_2\left(g\right):=\underset{t^{1/{\gamma }_2}<\left|u\right|}{\sup }\left\{{\displaystyle {\int }_{H^n}\left|K_{{\gamma }_2,t,\gamma }^{L,{\gamma }_1}\left(gu,h\right)-K_{{\gamma }_2,t,\gamma }^{L,{\gamma }_1}\left(g,h\right)\right|\left|b\left(gu\right)-b\left(h\right)\right|\left|f\left(h\right)\right|\text{d}h}\right\}.\end{array}$

对于 $I_1\left(g\right)$，若 $t^{1/{\gamma }_2}\ge \left|u\right|$，注意到 $\gamma \in \left(0,1\right)$ 和 $\left|u\right|\le \gamma /4$，通过(2)和(3)，则有

$\begin{array}{l}\left|K_{{\gamma }_2,t,\gamma }^{L,{\gamma }_1}\left(gu,h\right)-K_{{\gamma }_2,t,\gamma }^{L,{\gamma }_1}\left(g,h\right)\right|\\ \le \left|K_{{\gamma }_2,t}^{L,{\gamma }_1}\left(g,h\right)-K_{{\gamma }_2,t}^{L,{\gamma }_1}\left(gu,h\right)\right|\left|1+\phi \left({\gamma }^{-1}\left|{\left(gu\right)}^{-1}h\right|\right)\right|+\left|K_{{\gamma }_2,t}^{L,{\gamma }_1}\left(g,h\right)\right|\left|\phi \left({\gamma }^{-1}\left|g^{-1}h\right|\right)-\phi \left({\gamma }^{-1}\left|{\left(gu\right)}^{-1}h\right|\right)\right|\\ \le \frac{C_N{t}^{{\gamma }_1}}{{\left(t^{1/{\gamma }_2}+\left|g^{-1}h\right|\right)}^{\vartheta +{\gamma }_1{\gamma }_2}}\left({\left(\frac{\left|u\right|}{t^{1/{\gamma }_2}}\right)}^{\delta }+\left(\frac{\left|u\right|}{t^{1/{\gamma }_2}}\right)\right){\left(1+\frac{t^{1/{\gamma }_2}}{\rho \left(g\right)}+\frac{t^{1/{\gamma }_2}}{\rho \left(h\right)}\right)}^{-N},\end{array}$

其中 $\left|\phi \left({\gamma }^{-1}\left|g^{-1}h\right|\right)-\phi \left({\gamma }^{-1}\left|{\left(gu\right)}^{-1}h\right|\right)\right|\le C\left|u\right|/\gamma$。因此， $\left|I_1\left(g\right)\right|\le C\left\{I_{1,1}\left(g\right)+I_{1,2}\left(g\right)+I_{1,3}\left(g\right)+I_{1,4}\left(g\right)\right\}$，其中

若 $t^{1/{\gamma }_2}\ge 1$，有 ${\left(t^{1/{\gamma }_2}\right)}^{-\delta }\le 1$。对 $I_{1,1}\left(g\right)$，选取 $N=\theta \eta$，则有

$I_{1,1}\left(g\right)\le 2{\Vert b\Vert }_{L^{\infty }\left(H^n\right)}\underset{t^{1/{\gamma }_2}\ge 1}{\sup }\left\{\frac{\left({\left|u\right|}^{\delta }+\left|u\right|\right)}{\gamma {\left(t^{1/{\gamma }_2}\right)}^{\vartheta }}{\left(1+\frac{t^{1/{\gamma }_2}}{\rho \left(g\right)}\right)}^{-\theta \eta }{\displaystyle {\int }_{\left|g^{-1}h\right|<t^{1/{\gamma }_2}}\left|f\left(h\right)\right|\text{d}h}\right\}\le C{\gamma }^{-1}\left({\left|u\right|}^{\delta }+\left|u\right|\right)M_{V,\eta }\left(f\right)\left(g\right).$

注意到 ${\gamma }_1{\gamma }_2-\theta \eta >1>0$，类似于 $I_{1,1}\left(g\right)$ 能得到 $I_{1,2}\left(g\right)\le C{\gamma }^{-1}\left({\left|u\right|}^{\delta }+\left|u\right|\right)M_{V,\eta }\left(f\right)\left(g\right)$。

若 $t^{1/{\gamma }_2}<1$，有 ${\left(t^{1/{\gamma }_2}\right)}^{-\delta }\le {\left(t^{1/{\gamma }_2}\right)}^{-\left(1-\vartheta /q\right)}\le t^{-1/{\gamma }_2}$。对 $I_{1,3}\left(g\right)$，选取 $N=\theta \eta$，若 $\left|u\right|\le t^{1/{\gamma }_2}$， $\left|g^{-1}h\right|\le t^{1/{\gamma }_2}$ 且 $b\in C_c^{\infty }\left(H^n\right)$，则有 $\left|b\left(gu\right)-b\left(h\right)\right|\le C\left\{\left|g^{-1}h\right|+\left|u\right|\right\}\le C{t}^{1/{\gamma }_2}$。由 $\gamma \in \left(0,1\right)$ 能得到

$I_{1,3}\left(g\right)\le C\underset{\left|u\right|\le t^{1/{\gamma }_2}<1}{\sup }\left\{\left({\left|u\right|}^{\delta }+\frac{\left|u\right|t^{1/{\gamma }_2}}{\gamma }\right){\left(1+\frac{t^{1/{\gamma }_2}}{\rho \left(g\right)}\right)}^{-\theta \eta }\frac{1}{{\left(t^{1/{\gamma }_2}\right)}^{\vartheta }}{\displaystyle {\int }_{\left|g^{-1}h\right|<t^{1/{\gamma }_2}}\left|f\left(h\right)\right|\text{d}h}\right\}\le C{\gamma }^{-1}\left({\left|u\right|}^{\delta }+\left|u\right|\right)M_{V,\eta }\left(f\right)\left(g\right).$

对 $I_{1,4}\left(g\right)$，由于 $b\in C_c^{\infty }\left(H^n\right)$，若 $\left|g^{-1}h\right|~2^j{t}^{1/{\gamma }_2},j=1,2,\cdots$ 且 $\left|u\right|\le t^{1/{\gamma }_2}$，则有 $\left|b\left(gu\right)-b\left(h\right)\right|\le C\left\{\left|g^{-1}h\right|+\left|u\right|\right\}\le C{2}^j{t}^{1/{\gamma }_2}$。故而有

$\begin{array}{c}{I}_{1,4}\left(g\right)\le C\underset{\left|u\right|\le t^{1/{\gamma }_2}<1}{\sup }\left\{\left({\left|u\right|}^{\delta }+\frac{\left|u\right|t^{1/{\gamma }_2}}{\gamma }\right){\displaystyle \underset{j=1}{\overset{\infty }{\sum }}{\left(1+\frac{t^{1/{\gamma }_2}}{\rho \left(g\right)}\right)}^{-\theta \eta }{\displaystyle {\int }_{\left|g^{-1}h\right|\sim 2^j{t}^{1/{\gamma }_2}}\frac{t^{{\gamma }_1}{2}^j\left|f\left(h\right)\right|}{{\left(2^j{t}^{1/{\gamma }_2}\right)}^{{\gamma }_1{\gamma }_2}{\left(2^j{t}^{1/{\gamma }_2}\right)}^{\vartheta }}\text{d}h}}\right\}\\ \le C{\gamma }^{-1}\left({\left|u\right|}^{\delta }+\left|u\right|\right){\displaystyle \underset{j=1}{\overset{\infty }{\sum }}\frac{1}{{\left(2^j\right)}^{{\gamma }_1{\gamma }_2-1-\theta \eta }}}{M}_{V,\eta }\left(f\right)\left(g\right)\\ \le C{\gamma }^{-1}\left({\left|u\right|}^{\delta }+\left|u\right|\right)M_{V,\eta }\left(f\right)\left(g\right),\end{array}$

其中 ${\gamma }_1{\gamma }_2-\left(\theta \eta +1\right)>0$。这些 $I_{1,i}(\cdot ),i=1,2,3,4$ 的估计表明 $I_1\left(g\right)\le C{\gamma }^{-1}\left({\left|u\right|}^{\delta }+\left|u\right|\right)M_{V,\eta }\left(f\right)\left(g\right)$。

最后，估计 $I_2\left(g\right)$。若 $\left|g^{-1}h\right|<\gamma /2$ 和 $\left|u\right|<\gamma /4$，有 $\left|{\left(gu\right)}^{-1}h\right|<3\gamma /4$。因此 $\phi \left({\gamma }^{-1}\left|{\left(gu\right)}^{-1}h\right|\right)=\phi \left({\gamma }^{-1}\left|g^{-1}h\right|\right)=1$。这表明 $K_{{\gamma }_2,t,\gamma }^{L,{\gamma }_1}\left(gu,h\right)=K_{{\gamma }_2,t,\gamma }^{L,{\gamma }_1}\left(g,h\right)=0$，进而有 $I_2\left(g\right)=0$。故只需考虑 $\left|g^{-1}h\right|\ge \gamma /2$。

此时，有 $I_2\left(g\right)\le I_{2,1}\left(g\right)+I_{2,2}\left(g\right)$，其中

$\{\begin{array}{l}{I}_{2,1}\left(g\right):=\underset{t^{1/{\gamma }_2}<\left|u\right|<\rho \left(g\right)}{\sup }\left\{{\displaystyle {\int }_{\left|g^{-1}h\right|\ge \gamma /2}\left|K_{{\gamma }_2,t,\gamma }^{L,{\gamma }_1}\left(gu,h\right)-K_{{\gamma }_2,t,\gamma }^{L,{\gamma }_1}\left(g,h\right)\right|\left|b\left(gu\right)-b\left(h\right)\right|\left|f\left(h\right)\right|\text{d}h}\right\};\\ I_{2,2}\left(g\right):=\underset{\begin{array}{l}{t}^{1/{\gamma }_2}<\left|u\right|\\ \rho \left(g\right)\le \left|u\right|\end{array}}{\sup }\left\{{\displaystyle {\int }_{\left|g^{-1}h\right|<\gamma /2}\left|K_{{\gamma }_2,t,\gamma }^{L,{\gamma }_1}\left(gu,h\right)-K_{{\gamma }_2,t,\gamma }^{L,{\gamma }_1}\left(g,h\right)\right|\left|b\left(gu\right)-b\left(h\right)\right|\left|f\left(h\right)\right|\text{d}h}\right\}.\end{array}$

对 $I_{2,1}\left(g\right)$，由 $\left|g^{-1}h\right|>2\left|u\right|$，知 $\left|{\left(gu\right)}^{-1}h\right|~\left|g^{-1}h\right|$。由 $t^{1/{\gamma }_2}<\rho \left(g\right)$ 推出 $\rho \left(g\right)~\rho \left(gu\right)$， $\left|u\right|/\rho \left(g\right)<1$ 和 $\left|u\right|/t^{1/{\gamma }_2}>1$。进而有 $\left|u\right|/t^{1/{\gamma }_2}<{\left(\left|u\right|/t^{1/{\gamma }_2}\right)}^{{\gamma }_1{\gamma }_2}$。可得

$\begin{array}{c}{I}_{2,1}\left(g\right)\le C\underset{t^{1/{\gamma }_2}<\left|u\right|<\rho \left(g\right)}{\sup }\left\{{\displaystyle {\int }_{\left|g^{-1}h\right|\ge 2\left|u\right|}\frac{t^{{\gamma }_1}}{|g^{-1}h{|}^{\vartheta +{\gamma }_1{\gamma }_2-1}}{\left(\frac{\left|u\right|}{t^{1/{\gamma }_2}}\right)}^{{\gamma }_1{\gamma }_2}\left|f\left(h\right)\right|\text{d}h}\right\}\\ \le C\underset{t^{1/{\gamma }_2}<\left|u\right|<\rho \left(g\right)}{\sup }\left\{{\displaystyle \underset{j=2}{\overset{\infty }{\sum }}{\displaystyle {\int }_{\left|g^{-1}h\right|\sim 2^j\left|u\right|}\frac{{\left|u\right|}^{{\gamma }_1{\gamma }_2}}{{\left(2^j\left|u\right|\right)}^{\vartheta +{\gamma }_1{\gamma }_2-1}}{\left(1+\frac{2^j\left|u\right|}{\rho \left(g\right)}\right)}^{\theta \eta }{\left(1+\frac{2^j\left|u\right|}{\rho \left(g\right)}\right)}^{-\theta \eta }\left|f\left(h\right)\right|\text{d}h}}\right\}\\ \le C{\displaystyle \underset{j=2}{\overset{\infty }{\sum }}\frac{\left|u\right|}{{\left(2^j\right)}^{{\gamma }_1{\gamma }_2-1-\theta \eta }}}{M}_{V,\eta }\left(f\right)\left(g\right)\le C\left|u\right|M_{V,\eta }\left(f\right)\left(g\right),\end{array}$

其中 ${\gamma }_1{\gamma }_2>\theta \eta +1>0$。接下来，只剩估计 $I_{2,2}\left(g\right)$。由 $b\in C_c^{\infty }\left(H^n\right)$ 和 $\left|g^{-1}h\right|\ge \gamma /2>2\left|u\right|$，有 $\left|b\left(gu\right)-b\left(h\right)\right|\le C\left|g^{-1}h\right|$ 和 $\left|{\left(gu\right)}^{-1}h\right|~\left|g^{-1}h\right|$。另外，若 $\left|g^{-1}h\right|<2^l\rho \left(g\right),l=1,2,\cdots$，通过 ，有 $\rho \left(g\right)\le C{\left(1+\left|g^{-1}h\right|/\rho \left(g\right)\right)}^{k_0/\left(k_0+1\right)}\rho \left(g\right)\le C{2}^{k_{0}l/\left(k_0+1\right)}\rho \left(g\right)$。从而有

${\left(1+\frac{t^{1/{\gamma }_2}}{\rho \left(gu\right)}+\frac{t^{1/{\gamma }_2}}{\rho \left(h\right)}\right)}^{-N}+{\left(1+\frac{t^{1/{\gamma }_2}}{\rho \left(g\right)}+\frac{t^{1/{\gamma }_2}}{\rho \left(h\right)}\right)}^{-N}\le 2{\left(1+\frac{t^{1/{\gamma }_2}}{\rho \left(h\right)}\right)}^{-N}\le C{\left(1+\frac{2^{-k_{0}l/\left(k_0+1\right)}{t}^{1/{\gamma }_2}}{\rho \left(g\right)}\right)}^{-N}.$

取 $N={\gamma }_1{\gamma }_2-1$ 并应用 ，则有

$\begin{array}{c}{I}_{2,2}\left(g\right)\le C\underset{\begin{array}{l}{t}^{1/{\gamma }_2}<\left|u\right|\\ \rho \left(g\right)\le \left|u\right|\end{array}}{\sup }\left\{{\displaystyle {\int }_{\left|g^{-1}h\right|\ge 2\rho \left(g\right)}\frac{t^{{\gamma }_1}\left|u\right|/t^{1/{\gamma }_2}}{{\left|g^{-1}h\right|}^{\vartheta +{\gamma }_1{\gamma }_2-1}}{\left(1+\frac{2^{-k_{0}l/\left(k_0+1\right)}{t}^{1/{\gamma }_2}}{\rho \left(g\right)}\right)}^{-N}\left|f\left(h\right)\right|\text{d}h}\right\}\\ \le C\underset{\begin{array}{l}{t}^{1/{\gamma }_2}<\left|u\right|\\ \rho \left(g\right)\le \left|u\right|\end{array}}{\sup }\left\{{\displaystyle \underset{l=2}{\overset{\infty }{\sum }}\frac{\left|u\right|{\left(2^{-k_{0}l/\left(k_0+1\right)}\right)}^{1-{\gamma }_1{\gamma }_2}}{{\left(2^l\right)}^{{\gamma }_1{\gamma }_2-1-\theta \eta }{\left(2^l\rho \left(g\right)\right)}^{\vartheta }}{\left(1+\frac{2^l\rho \left(g\right)}{\rho \left(g\right)}\right)}^{-\theta \eta }{\displaystyle {\int }_{\left|g^{-1}h\right|\sim 2^l\rho \left(g\right)}\left|f\left(h\right)\right|\text{d}h}}\right\}\\ \le C\left|u\right|{\displaystyle \underset{l=2}{\overset{\infty }{\sum }}\frac{1}{{\left(2^l\right)}^{\left({\gamma }_1{\gamma }_2-1\right)/\left(k_0+1\right)-\theta \eta }}{M}_{V,\eta }\left(f\right)\left(g\right)}\le C\left|u\right|M_{V,\eta }\left(f\right)\left(g\right),\end{array}$

其中 ${\gamma }_1{\gamma }_2-1>\left(1+k_0\right)\theta \eta$。 $I_{2,1}\left(g\right)$ 和 $I_{2,2}\left(g\right)$ 的估计表明 $I_2\left(g\right)\le C\left|u\right|M_{V,\eta }\left(f\right)\left(g\right)$。结合 $I_1\left(g\right)$ 和 $I_2\left(g\right)$ 的估计，能得到 $I\left(g\right)\le C\left({\left|u\right|}^{\delta }+\left|u\right|\right)M_{V,\eta }\left(f\right)\left(g\right)$。由引理2.4，其中 $p^{\prime }\le \eta <\infty$，有

${\Vert I\Vert }_{L^p\left(H^n\right)}\le C\left({\left|u\right|}^{\delta }+\left|u\right|\right){\Vert M_{V,\eta }\left(f\right)\Vert }_{L^p\left(H^n\right)}\le C\left({\left|u\right|}^{\delta }+\left|u\right|\right){\Vert f\Vert }_{L^p\left(H^n\right)}.$

通过 $I\left(g\right)$ 和 $II\left(g\right)$ 的估计，有 ${\Vert \left[b,{\Phi }_{{\gamma }_1,{\gamma }_2,\gamma }^{L,*}\right]\left(f\right)\left(\cdot u\right)-\left[b,{\Phi }_{{\gamma }_1,{\gamma }_2,\gamma }^{L,*}\right]\left(f\right)(\cdot )\Vert }_{L^p\left(H^n\right)}\le C\left({\left|u\right|}^{\delta }+\left|u\right|\right){\Vert f\Vert }_{L^p\left(H^n\right)}$ 成立。

因此，我们得到了(7)并完成了证明。

定理4.2假设 $V\in B_q,q>\vartheta /2$， $b\in CMO\left(\rho \right)\left(H^n\right)$ 且 $\theta >0$，则 $\left[b,T_i^{L,*}\right],i=1,2$ 是 $L^p\left(H^n\right)$ 上的一个紧算子。

证明首先，通过引理2.8，当 ${\gamma }_2=2\alpha$ 且 ${\gamma }_1=m$ 时，有 $t^m{\partial }_t^m{K}_{\alpha ,t}^L\left(g,h\right)$ 满足条件(2)和(3)。因此，通过定理4.1，得出 $\left[b,T_1^{L,*}\right]$ 是 $L^p\left(H^n\right)$ 上的一个紧算子。其次，通过定理4.1，仅需证 $t^{\upsilon }{\partial }_t^{\upsilon }{p}_{t,\upsilon }^{L,\alpha }\left(g,h\right)$ 满足条件(2)

和(3)。先证 $t^{\upsilon }{\partial }_t^{\upsilon }{p}_{t,\upsilon }^{L,\alpha }\left(g,h\right)$ 满足条件(2)。由于 $t^{\upsilon }{\partial }_t^{\upsilon }{p}_{t,\upsilon }^{L,\alpha }\left(g,h\right)=C{\displaystyle {\int }_0^{\infty }{\text{e}}^{-r}{t}^{\upsilon }{\partial }_t^{\upsilon }{K}_{\alpha ,t^{2\alpha }/\left(4r\right)}^L\left(g,h\right)}\text{d}r/r^{1-\sigma }$，通过

引理2.8的(1)，对 $L>\vartheta /\left(2\alpha \right)+N/\alpha +\sigma >0$，有

$\begin{array}{c}{t}^{\upsilon }{\partial }_t^{\upsilon }{p}_{t,\upsilon }^{L,\alpha }\left(g,h\right)\le C{\displaystyle {\int }_0^{\infty }{\text{e}}^{-r}{\left(t^{2\alpha }/\left(4r\right)\right)}^{-\vartheta /\left(2\alpha \right)}{\left(t^{2\alpha }/\left(4r\right)\right)}^{-N/\alpha }\rho {\left(g\right)}^N}\rho {\left(h\right)}^N\frac{\text{d}r}{r^{1-\sigma }}\\ \le \frac{C}{t^{\vartheta +2N}}\rho {\left(g\right)}^N\rho {\left(h\right)}^N\left({\displaystyle {\int }_0^1}+{\displaystyle {\int }_1^{\infty }}\right){\text{e}}^{-r}{r}^{\vartheta /\left(2\alpha \right)+N/\alpha +\sigma -1}\text{d}r\\ \le C{\left(\frac{t}{\rho \left(g\right)}\right)}^{-N}{\left(\frac{t}{\rho \left(h\right)}\right)}^{-N}\frac{1}{t^{\vartheta }}.\end{array}$

另一方面，选取 $L>\alpha /N-\upsilon +\sigma >0$，有

$t^{\upsilon }{\partial }_t^{\upsilon }{p}_{t,\upsilon }^{L,\alpha }\left(g,h\right)\le C\rho {\left(g\right)}^N\rho {\left(h\right)}^N\frac{t^{2\alpha \upsilon -2N}}{{\left|g^{-1}h\right|}^{\vartheta +2\alpha \upsilon }}{\displaystyle {\int }_0^{\infty }{\text{e}}^{-r}{r}^{\alpha /N-\upsilon +\sigma -1}}\text{d}r\le C{\left(\frac{t}{\rho \left(g\right)}\right)}^{-N}{\left(\frac{t}{\rho \left(h\right)}\right)}^{-N}\frac{t^{2\alpha \upsilon }}{{\left|g^{-1}h\right|}^{\vartheta +2\alpha \upsilon }}.$

考虑两种情况： $0\le t\le \left|g^{-1}h\right|$ 和 $t>\left|g^{-1}h\right|$。显然地，都能得到

${\left(\frac{t}{\rho \left(g\right)}\right)}^N{\left(\frac{t}{\rho \left(h\right)}\right)}^N\left|t^{\upsilon }{\partial }_t^{\upsilon }{p}_{t,\upsilon }^{L,\alpha }\left(g,h\right)\right|\le C\frac{t^{2\alpha \upsilon }}{{\left(t+\left|g^{-1}h\right|\right)}^{\vartheta +2\alpha \upsilon }}.$

从而有 $t^{\upsilon }{\partial }_t^{\upsilon }{p}_{t,\upsilon }^{L,\alpha }\left(g,h\right)$ 满足条件(2)，其中 ${\gamma }_2=1$ 且 ${\gamma }_1=2\alpha \upsilon$。 $t^{\upsilon }{\partial }_t^{\upsilon }{p}_{t,\upsilon }^{L,\alpha }\left(g,h\right)$ 满足条件(3)的证明相似于 $p_{t,\upsilon }^{L,\alpha }\left(g,h\right)$，此处省略细节。故得出 $\left[b,T_2^{L,*}\right]$ 是 $L^p\left(H^n\right)$ 上的一个紧算子。

致谢

作者衷心感谢李澎涛教授的指导与建议。

基金项目

山东省自然科学基金(项目编号：ZR2017JL008)。