伊犁师范大学数学与统计学院，新疆 伊宁

收稿日期：2022年11月25日；录用日期：2022年12月21日；发布日期：2022年12月30日

摘要

本文考虑次线性算子的多线性交换子在变指标Herz Triebel-Lizorkin空间的有界性问题。通过利用变指标Herz Triebel-Lizorkin空间的等价刻画，极大算子控制法及Lipschitz函数的相关性质，证明了次线性算子与Lipschitz函数生成的多线性交换子是从变指标Herz空间到变指标Herz Triebel-Lizorkin空间有界的。

关键词

次线性算子，多线性交换子，极大算子，变指标Herz Triebel-Lizorkin空间

Boundedness of Multilinear Commutators for Sublinear Operators on the Herz Triebel-Lizorkin Spaces with Variable Exponent

Wanjing Zhang, Xin Cheng

College of Mathematics and Statistics, Yili Normal University, Yining Xinjiang

Received: Nov. 25^th, 2022; accepted: Dec. 21^st, 2022; published: Dec. 30^th, 2022

ABSTRACT

This paper is concerned with a boundedness problem for the multilinear commutators of sublinear operators on the Herz Triebel-Lizorkin spaces with variable exponent. By means of the maximum operator control method, the equivalent characterized of Herz Triebel-Lizorkin spaces with variable exponent and related properties of Lipschitz functions, it proved the boundedness of multilinear commutators for sublinear operators with Lipschitz functions from variable exponent Herz spaces to variable exponent Herz Triebel-Lizorkin spaces.

Keywords:Sublinear Operators, Multilinear Commutators, Maximum Operator, Herz Triebel-Lizorkin Spaces with Variable Exponent

Copyright © 2022 by author(s) and Hans Publishers Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).

http://creativecommons.org/licenses/by/4.0/

1. 引言

近年来，自1991年Kováčik和Rákosník [1] 给出了变指标Lebesgue空间和Sobolev空间的完备性，连续性及等价范数后，变指标函数空间得到了迅速的发展。众所周知，Herz空间是Lebesgue空间的加幂权形式。随着2010年Izuki [2] 引入了变指标Herz空间并研究了该空间的性质，建立了次线性算子在该空间的有界性之后，越来越多的学者开始关注并研究变指标Herz型空间。先后定义了变指标Herz-Morrey空间 [3]、变指标Herz-Hardy空间 [4]、变指标Herz-Besov空间 [5] 和变指标Herz Triebel-Lizorkin空间 [5]。最近，韦营营等 [6]，Fang等 [7] 利用Peetre极大算子和Hardy-Littlewood极大算子在向量值空间的有界性，建立了变指标Herz Triebel-Lizorkin空间的等价范数刻画。

另一方面，关于交换子的研究也取得了很大进展。如2015年，王洪斌 [8] 证明了次线性算子的交换子在变指标Herz-Hardy空间的有界性。2017年，王立伟等 [9] 得到了次线性算子的多线性交换子在变指标Herz空间的有界性结果。2022年，彭珊珊等 [10] 证明了Calderón-Zygmund算子的多线性交换子在极大变指标Herz空间的有界性。受文献 [6] 和文献 [9] 研究结果的启发，本文主要讨论次线性算子与Lipschitz函数生成的多线性交换子在变指标Herz Triebel-Lizorkin空间的有界性问题，对变指标函数空间理论做了进一步的推广。

本文结构安排如下：将在第二节给出与本文相关的定义、引理，在第三节给出本文的主要结果与相关证明。

2. 预备知识

在这篇文章中，我们引入次线性算子T和 $T_{\lambda }$ 与局部可积函数 $b$ 构成的多线性交换子 $T^b$ 和 $T_{\lambda }^b$，$T^b$ 和 $T_{\lambda }^b$ 分别定义如下：

$T^b\left(f\right)\left(x\right)=\Vert F_t^b\left(f\right)\left(x\right)\Vert$，$T_{\lambda }^b\left(f\right)\left(x\right)=\Vert E_t^b\left(f\right)\left(x\right)\Vert$，

在这里，假设 $b_j\left(j=1,\cdots ,m\right)$ 是固定在 ${ℝ}^n$ 上的局部可积函数， $F_t\left(x,y\right)$,$E_t\left(x,y\right)$ 都定义在 ${ℝ}^n\times {ℝ}^n\times \left[0,+\infty \right)$ 上，

$F_t\left(f\right)\left(x\right)={\displaystyle {\int }_{{ℝ}^n}{F}_t}\left(x,y\right)f\left(y\right)\text{d}y$，$E_t\left(f\right)\left(x\right)={\displaystyle {\int }_{{ℝ}^n}{E}_t\left(x,y\right)f\left(y\right)\text{d}y}$，

且

$F_t^b\left(f\right)\left(x\right)={\displaystyle {\int }_{{ℝ}^n}{\displaystyle \underset{j=1}{\overset{m}{\prod }}\left(b_j\left(x\right)-b_j\left(y\right)\right))F_t\left(x,y\right)f\left(y\right)\text{d}y}}$，$E_t^b\left(f\right)\left(x\right)={\displaystyle {\int }_{{ℝ}^n}{\displaystyle \underset{j=1}{\overset{m}{\prod }}\left(b_j\left(x\right)-b_j\left(y\right)\right)E_t\left(x,y\right)f\left(y\right)\text{d}y}}$。

$F_t$，$E_t$ 分别满足以下条件：对给定的 $\epsilon >0$，$0<\lambda <n$

$\Vert F_t\left(x,y\right)\Vert \le C{\left|x-y\right|}^{-n}$,(1)

$\Vert E_t\left(x,y\right)\Vert \le C{\left|x-y\right|}^{-n+\lambda }$， (2)

和当 $2\left|y-z\right|\le \left|x-z\right|$ 时，

$\Vert F_t\left(y,x\right)-F_t\left(z,x\right)\Vert +\Vert F_t\left(x,y\right)-F_t\left(x,z\right)\Vert \le C{\left|y-z\right|}^{\epsilon }{\left|x-z\right|}^{-n-\epsilon }$，

$\Vert E_t\left(y,x\right)-E_t\left(z,x\right)\Vert \le C{\left|y-z\right|}^{\epsilon }{\left|x-z\right|}^{-n-\epsilon +\lambda }$。

同样的，也可以定义 $T\left(f\right)\left(x\right)=\Vert F_t\left(f\right)\left(x\right)\Vert$，$T_{\lambda }\left(f\right)\left(x\right)=\Vert E_t\left(f\right)\left(x\right)\Vert$。

对于变指标Herz Triebel-Lizorkin空间，有如下重要性质：

对 $p(\cdot )\in \mathcal{B}\left({ℝ}^n\right)$，$0<\beta <1$，$\alpha \in ℝ$，$0<q\le \infty$，齐次变指标Herz Triebel-Lizorkin空间的等价范数刻画如下：

${\Vert f\Vert }_{{\stackrel{\dot{}}{K}}_{p(\cdot )}^{\alpha ,q}{\stackrel{\dot{}}{F}}_{\infty }^{\beta }\left({ℝ}^n\right)}\sim {\Vert \underset{Q}{\sup }\frac{1}{{\left|Q\right|}^{1+\beta /n}}{\displaystyle {\int }_Q\left|f-f_Q\right|}\Vert }_{{\stackrel{\dot{}}{K}}_{p(\cdot )}^{\alpha ,q}\left({ℝ}^n\right)}$，

非齐次变指标Herz Triebel-Lizorkin空间的等价范数刻画如下：

${\Vert f\Vert }_{K_{p(\cdot )}^{\alpha ,q}{\stackrel{\dot{}}{F}}_{\infty }^{\beta }\left({ℝ}^n\right)}\sim {\Vert \underset{Q}{\sup }\frac{1}{{\left|Q\right|}^{1+\beta /n}}{\displaystyle {\int }_Q\left|f-f_Q\right|}\Vert }_{K_{p(\cdot )}^{\alpha ,q}\left({ℝ}^n\right)}$，

其中 $f_Q=\frac{1}{\left|Q\right|}{\displaystyle {\int }_{Q}f\left(x\right)\text{d}x}$。

下面介绍一些定义、引理：

定义1 [6] 设 $\Omega$ 是在 ${ℝ}^n$ 上的一个可测子集且 $\left|\Omega \right|>0$。令 $p(\cdot ):\Omega \to \left[1,\infty \right)$ 是可测函数。变指标Lebesgue空间 $L^{p(\cdot )}\left(\Omega \right)$ 定义为：对于某个 $\lambda >0$，使得

${\displaystyle {\int }_{\Omega }\left|\frac{f\left(x\right)}{\lambda }\right|\text{d}x}<\infty$,

成立的 $\Omega$ 上的可测函数f全体。其范数可以表示为：

${\Vert f\Vert }_{L^{p(\cdot )}\left(\Omega \right)}:=\inf \left\{\lambda >0:{\displaystyle {\int }_{\Omega }{\left|\frac{f\left(x\right)}{\lambda }\right|}^{p\left(x\right)}\text{d}x}\le 1\right\}$。

设 $L_{loc}^1\left({ℝ}^n\right)$ 是 ${ℝ}^n$ 上所有局部可积函数的集合，给定一个函数 $f\in L_{loc}^1\left({ℝ}^n\right)$，则Hardy-Littlewood极大算子M被定义为：

$Mf\left(x\right):=\underset{r>0}{\sup }{r}^{-n}{\displaystyle {\int }_{B\left(x,r\right)}\left|f\left(y\right)\right|\text{d}y}$，$\forall x\in {ℝ}^n$，

这里 $B\left(x,r\right)=\left\{y\in {ℝ}^n:\left|x-y\right|<r\right\}$。

令 $p_{-}:=\mathrm{ess}\inf \left\{p\left(x\right):x\in {ℝ}^n\right\}>0$，$p_{+}:=\mathrm{ess}\sup \left\{p\left(x\right):x\in {ℝ}^n\right\}<\infty$，记 ${\mathcal{P}}^0\left({ℝ}^n\right)$ 是满足 $0<p_{-}\le p\left(x\right)<p_{+}<\infty$ 的所有 $p\left(x\right)$ 的全体， $\mathcal{P}\left({ℝ}^n\right)$ 是满足 $1<p_{-}\le p\left(x\right)<p_{+}<\infty$ 的所有 $p\left(x\right)$ 的全体。

$p^{\prime }(\cdot )$ 是 $p(\cdot )$ 的共轭指数，即 $p^{\prime }(\cdot )=\frac{p(\cdot )}{p(\cdot )-1}$。用 $\mathcal{B}\left({ℝ}^n\right)$ 表示满足 $p(\cdot )\in \mathcal{P}\left({ℝ}^n\right)$ 且使得Hardy-Littlewood极大算子M在 $L^{p(\cdot )}\left({ℝ}^n\right)$ 上有界的全体。若 $p(\cdot )\in \mathcal{P}\left({ℝ}^n\right)$ 满足

$\left|p\left(x\right)-p\left(y\right)\right|\le \frac{-C}{\log \left(\left|x-y\right|\right)}$，$\left|x-y\right|\le \frac{1}{2}$，

$\left|p\left(x\right)-p\left(y\right)\right|\le \frac{C}{\log \left(e+\left|x\right|\right)}$，$\left|y\right|\ge \left|x\right|$，

则 $p(\cdot )\in \mathcal{B}\left({ℝ}^n\right)$。

接下来给出一些记号以及Lipschitz函数的相关性质：

对 $1\le l\le m$，$C_l^m$ 表示由 $\left\{1,2,\cdots ,m\right\}$ 中l个不同元素构成的有限子集族 $\sigma =\left\{\sigma \left(1\right),\cdots ,\sigma \left(l\right)\right\}$。对任意 $\sigma \in C_l^m$，用 ${\sigma }^c=\left\{1,\cdots ,m\right\}\backslash \sigma$ 表示 $\sigma$ 的补序列。如果 $b=\left(b_1,b_2,\cdots ,b_m\right)$，$b_l\in {\mathrm{Lip}}_{\beta }\left({ℝ}^n\right)$ 及

$\sigma =\left\{\sigma \left(1\right),\cdots ,\sigma \left(l\right)\right\}\in C_l^m$，$1\le l\le m$，记 $b_{\sigma }=b_{\sigma \left(1\right)}\cdots b_{\sigma \left(l\right)}$，${\left(b\left(x\right)-b_B\right)}_{\sigma }:={\displaystyle \underset{i=1}{\overset{l}{\prod }}\left(\left(b_{\sigma \left(i\right)}\right)\left(x\right)-{\left(b_{\sigma \left(i\right)}\right)}_B\right)}$。并记 ${\Vert b_{\sigma }\Vert }_{{\mathrm{Lip}}_{\beta }\left({ℝ}^n\right)}:={\displaystyle \underset{i=1}{\overset{l}{\prod }}{\Vert b_{\sigma \left(i\right)}\Vert }_{{\mathrm{Lip}}_{\beta }\left({ℝ}^n\right)}}$。特别地， ${\Vert b\Vert }_{{\mathrm{Lip}}_{\beta }\left({ℝ}^n\right)}:={\displaystyle \underset{l=1}{\overset{m}{\prod }}{\Vert b_l\Vert }_{{\mathrm{Lip}}_{\beta }\left({ℝ}^n\right)}}$。

引理1 [11] 对 $0<\beta <1$，$1<q\le \infty$，有

${\Vert f\Vert }_{{\mathrm{Lip}}_{\beta }\left({ℝ}^n\right)}=\underset{Q}{\sup }\frac{1}{{\left|Q\right|}^{1+\beta /n}}{\displaystyle {\int }_Q\left|f\left(y\right)-f_Q\right|\text{d}y}\approx \underset{Q}{\sup }\frac{1}{{\left|Q\right|}^{\beta /n}}{\left(\frac{1}{\left|Q\right|}{\displaystyle {\int }_Q{\left|f\left(y\right)-f_Q\right|}^q\text{d}y}\right)}^{1/q}$，

当 $q=\infty$ 时，上式做相应修改。

引理2 [12] 设 $Q^{\text{*}}\subset Q$，则 $\left|f_{Q^{*}}-f_Q\right|\le {\Vert f\Vert }_{{\mathrm{Lip}}_{\beta }\left({ℝ}^n\right)}{\left|Q\right|}^{\beta /n}$。

目前，关于极大算子和次线性算子在变指标Herz空间的有界性结果有：

引理3 [5] 令 $p(\cdot )\in \mathcal{B}\left({ℝ}^n\right)$，$0<q<\infty$，$1<r<\infty$。若 ${\delta }_1,{\delta }_2\in \left(0,1\right)$ 且 $-n{\delta }_1<\alpha <n{\delta }_2$。则存在一个正常数C使得对 ${ℝ}^n$ 上所有局部可积的函数序列 ${\left\{f_j\right\}}_{j=-\infty }^{\infty }$，有

${\Vert {\left({\displaystyle \underset{k=-\infty }{\overset{\infty }{\sum }}{\left|M{f}_k\right|}^r}\right)}^{1/r}\Vert }_{{\stackrel{\dot{}}{K}}_{p(\cdot )}^{\alpha \cdot q}\left({ℝ}^n\right)}\le C{\Vert {\left({\displaystyle \underset{k=-\infty }{\overset{\infty }{\sum }}{\left|f_k\right|}^r}\right)}^{1/r}\Vert }_{{\stackrel{\dot{}}{K}}_{p(\cdot )}^{\alpha \cdot q}\left({ℝ}^n\right)}$。

引理4 [2] 令 $\alpha \in ℝ$，$0<q<\infty$ 且 $p(\cdot )\in \mathcal{B}\left({ℝ}^n\right)$。若 ${\delta }_1,{\delta }_2\in \left(0,1\right)$ 且 $-n{\delta }_1<\alpha <n{\delta }_2$。假设T是一个次线性算子，满足尺寸条件(1)。若T在 $L^{p(\cdot )}\left({ℝ}^n\right)$ 上有界，那么T在变指标Herz空间有界。

引理5 [7] 令 $\alpha \in ℝ$，$0<\lambda <n$，$0<q_1<q_2\le \infty$，$p_1(\cdot ),p_2(\cdot )\in \mathcal{B}\left({ℝ}^n\right)$ 且 $\frac{1}{p_2(\cdot )}=\frac{1}{p_1(\cdot )}-\frac{\lambda }{n}$。若 ${\delta }_1,{\delta }_2\in \left(0,1\right)$ 且 $-n{\delta }_1<\alpha <n{\delta }_2$。假设 $T_{\lambda }$ 是一个次线性算子，满足尺寸条件(2)。若 $T_{\lambda }$ 从 $L^{p_1(\cdot )}\left({ℝ}^n\right)$ 到 $L^{p_2(\cdot )}\left({ℝ}^n\right)$ 有界，那么 $T_{\lambda }$ 是从齐次变指标Herz空间 ${\stackrel{\dot{}}{K}}_{p_1(\cdot )}^{\alpha ,q_1}\left({ℝ}^n\right)$ 到齐次变指标Herz空间 ${\stackrel{\dot{}}{K}}_{p_2(\cdot )}^{\alpha ,q_2}\left({ℝ}^n\right)$ 有界的，并且 $T_{\lambda }$ 在非齐次变指标Herz空间的有界性也同样成立。

引理6 [7] 令 $p_1(\cdot ),p_2(\cdot )\in \mathcal{B}\left({ℝ}^n\right)$ 且 $\frac{1}{p_1(\cdot )}-\frac{1}{p_2(\cdot )}=\frac{\lambda }{n}$。若 $h^Q$ 是固定在方体Q上的函数，那么对于 $0\le \beta$，有

${\Vert \underset{Q}{\sup }\frac{1}{{\left|Q\right|}^{1+\beta /n}}{\displaystyle {\int }_Q\left|h^Q\right|}\Vert }_{{\stackrel{\dot{}}{K}}_{p_2(\cdot )}^{\alpha ,q}\left({ℝ}^n\right)}\le C{\Vert \underset{Q}{\sup }\frac{1}{{\left|Q\right|}^{1+\beta /n+\lambda /n}}{\displaystyle {\int }_Q\left|h^Q\right|}\Vert }_{{\stackrel{\dot{}}{K}}_{p_1(\cdot )}^{\alpha ,q}\left({ℝ}^n\right)}$，

且

${\Vert \underset{Q}{\sup }\frac{1}{{\left|Q\right|}^{1+\beta /n}}{\displaystyle {\int }_Q\left|h^Q\right|}\Vert }_{K_{p_2(\cdot )}^{\alpha ,q}\left({ℝ}^n\right)}\le C{\Vert \underset{Q}{\sup }\frac{1}{{\left|Q\right|}^{1+\beta /n+\lambda /n}}{\displaystyle {\int }_Q\left|h^Q\right|}\Vert }_{K_{p_1(\cdot )}^{\alpha ,q}\left({ℝ}^n\right)}$，

这里的常数C仅仅与 $p_1(\cdot ),p_2(\cdot ),q,\alpha$ 和n有关。

3. 主要结果及证明

本文的主要结论如下：

定理1 令 $\alpha \in ℝ$，$0<q\le \infty$ 且 $p(\cdot )\in \mathcal{B}\left({ℝ}^n\right)$。假设 ${\delta }_1,{\delta }_2\in \left(0,1\right)$，$-n{\delta }_1<\alpha <n{\delta }_2$ 且 $0<\beta <\min \left(1,\frac{\epsilon }{m}\right)$，$b=\left(b_1,\cdots ,b_m\right)$，$b_j\in {\mathrm{Lip}}_{\beta }\left({ℝ}^n\right)$，$1\le j\le m$。对任意的具有紧支集的局部可积函数f，若次线性算子T满足尺寸条件(1)且T在 $L^{p(\cdot )}\left({ℝ}^n\right)$ 上有界，那么

(a) $T^b$ 是从 ${\stackrel{\dot{}}{K}}_{p(\cdot )}^{\alpha ,q}\left({ℝ}^n\right)$ 到 ${\stackrel{\dot{}}{K}}_{p(\cdot )}^{\alpha ,q}{\stackrel{\dot{}}{F}}_{\infty }^{m\beta }\left({ℝ}^n\right)$ 有界的；

(b) $T^b$ 是从 $K_{p(\cdot )}^{\alpha ,q}\left({ℝ}^n\right)$ 到 $K_{p(\cdot )}^{\alpha ,q}{\stackrel{\dot{}}{F}}_{\infty }^{m\beta }\left({ℝ}^n\right)$ 有界的。

定理2 令 $\alpha \in ℝ$，$0<\lambda <n$，$0<q_1,q_2<\infty$，$p_1(\cdot ),p_2(\cdot )\in \mathcal{B}\left({ℝ}^n\right)$ 且 $\frac{1}{p_2(\cdot )}=\frac{1}{p_1(\cdot )}-\frac{\lambda }{n}$。假设 ${\delta }_1,{\delta }_2\in \left(0,1\right)$，$-n{\delta }_1<\alpha <n{\delta }_2$ 且 $0<\beta <\min \left(1,\frac{\epsilon }{m}\right)$，$b=\left(b_1,\cdots ,b_m\right)$，$b_j\in {\mathrm{Lip}}_{\beta }\left({ℝ}^n\right)$，$1\le j\le m$。对任意的具有紧支集的局部可积函数f，若次线性算子 $T_{\lambda }$ 满足尺寸条件(2)且 $T_{\lambda }$ 从 $L^{p_1(\cdot )}\left({ℝ}^n\right)$ 到 $L^{p_2(\cdot )}\left({ℝ}^n\right)$ 有界，那么

(a) $T_{\lambda }^b$ 是从 ${\stackrel{\dot{}}{K}}_{p_1(\cdot )}^{\alpha ,q_1}\left({ℝ}^n\right)$ 到 ${\stackrel{\dot{}}{K}}_{p_2(\cdot )}^{\alpha ,q_2}{\stackrel{\dot{}}{F}}_{\infty }^{m\beta }\left({ℝ}^n\right)$ ；

(b) $T_{\lambda }^b$ 是从 $K_{p_1(\cdot )}^{\alpha ,q_1}\left({ℝ}^n\right)$ 到 $K_{p_2(\cdot )}^{\alpha ,q_2}{\stackrel{\dot{}}{F}}_{\infty }^{m\beta }\left({ℝ}^n\right)$。

定理1的证明：令 $Q=Q\left(x_0,l\right)$，$x_0\in Q$，$l>0$。对 $f\in L^{p(\cdot )}\left({ℝ}^n\right)$ 做如下分解： $f_1=f{\chi }_{2Q}$，$f_2=f-f_1$。有：

$\begin{array}{c}{F}_t^b\left(f\right)\left(x\right)={\displaystyle {\int }_{{ℝ}^n}{\displaystyle \underset{j=1}{\overset{m}{\prod }}\left(b_j\left(x\right)-b_j\left(y\right)\right)F_t\left(x,y\right)f\left(y\right)\text{d}y}}\\ ={\displaystyle {\int }_{{ℝ}^n}{\displaystyle \underset{j=1}{\overset{m}{\prod }}\left(b_j\left(x\right)-{\left(b_j\right)}_Q+{\left(b_j\right)}_Q-b_j\left(y\right)\right)F_t\left(x,y\right)f\left(y\right)\text{d}y}}\\ ={\displaystyle \underset{j=0}{\overset{m}{\sum }}{\displaystyle \underset{\sigma \in C_j^m}{\sum }{\left(-1\right)}^{m-j}{\left(b_j\left(x\right)-{\left(b_j\right)}_Q\right)}_{\sigma }}}{\displaystyle {\int }_{{ℝ}^n}{\left(b_j\left(y\right)-{\left(b_j\right)}_Q\right)}_{{\sigma }^c}{F}_t\left(x,y\right)f\left(y\right)\text{d}y}\end{array}$

$\begin{array}{c}=\left(b_1\left(x\right)-{\left(b_1\right)}_Q\right)\cdots \left(b_m\left(x\right)-{\left(b_m\right)}_Q\right)F_t\left(f\right)\left(x\right)\\ +{\left(-1\right)}^m{F}_t\left(\left(b_1-{\left(b_1\right)}_Q\right)\cdots \left(b_m-{\left(b_m\right)}_Q\right)f\right)\left(x\right)\\ +{\displaystyle \underset{j=1}{\overset{m-1}{\sum }}{\displaystyle \underset{\sigma \in C_j^m}{\sum }{\left(-1\right)}^{m-j}{\left(b\left(x\right)-b_Q\right)}_{\sigma }{\displaystyle {\int }_{{ℝ}^n}{\left(b\left(y\right)-b_Q\right)}_{{\sigma }^c}}}}{F}_t\left(x,y\right)f\left(y\right)\text{d}y\end{array}$

$\begin{array}{l}=\left(b_1\left(x\right)-{\left(b_1\right)}_Q\right)\cdots \left(b_m\left(x\right)-{\left(b_m\right)}_Q\right)F_t\left(f\right)\left(x\right)\\ +{\left(-1\right)}^m{F}_t\left(\left(b_1-{\left(b_1\right)}_Q\right)\cdots \left(b_m-{\left(b_m\right)}_Q\right)f_1\right)\left(x\right)\\ +{\left(-1\right)}^m{F}_t\left(\left(b_1-{\left(b_1\right)}_Q\right)\cdots \left(b_m-{\left(b_m\right)}_Q\right)f_2\right)\left(x\right)\\ +{\displaystyle \underset{j=1}{\overset{m-1}{\sum }}{\displaystyle \underset{\sigma \in C_j^m}{\sum }{\left(-1\right)}^{m-j}{\left(b\left(x\right)-b_Q\right)}_{\sigma }}}{F}_t\left({\left(b-b_Q\right)}_{{\sigma }^c}f\right)\left(x\right).\end{array}$

因此

$\begin{array}{l}\left|T^b\left(f\right)\left(x\right)-T\left(\left({\left(b_1\right)}_Q-b_1\right)\cdots \left({\left(b_m\right)}_Q-b_m\right)f_2\right)\left(x_0\right)\right|\\ \le \Vert F_t^b\left(f\right)\left(x\right)-F_t\left(\left({\left(b_1\right)}_Q-b_1\right)\cdots \left({\left(b_m\right)}_Q-b_m\right)f_2\right)\left(x_0\right)\Vert \\ \le \Vert \left(b_1\left(x\right)-{\left(b_1\right)}_Q\right)\cdots \left(b_m\left(x\right)-{\left(b_m\right)}_Q\right)F_t\left(f\right)\left(x\right)\Vert \\ +{\displaystyle \underset{j=1}{\overset{m-1}{\sum }}{\displaystyle \underset{\sigma \in C_j^m}{\sum }\Vert {\left(b\left(x\right)-b_Q\right)}_{\sigma }{F}_t\left({\left(b-b_Q\right)}_{{\sigma }^c}f\right)\left(x\right)\Vert }}\end{array}$

$\begin{array}{l}+\Vert F_t\left(\left(b_1-{\left(b_1\right)}_Q\right)\cdots \left(b_m-{\left(b_m\right)}_Q\right)f_1\right)\left(x\right)\Vert \\ +\Vert F_t\left({\displaystyle \underset{j=1}{\overset{m}{\prod }}\left(b_j-{\left(b_j\right)}_Q\right)f_2}\right)\left(x\right)-F_t\left({\displaystyle \underset{j=1}{\overset{m}{\prod }}\left(b_j-{\left(b_j\right)}_Q\right)f_2}\right)\left(x_0\right)\Vert \\ :=E_1\left(x\right)+E_2\left(x\right)+E_3\left(x\right)+E_4\left(x\right).\end{array}$

那么

$\begin{array}{l}\frac{1}{{\left|Q\right|}^{1+m\beta /n}}{\displaystyle {\int }_Q\left|T^b\left(f\right)\left(x\right)-T\left(\left({\left(b_1\right)}_Q-b_1\right)\cdots \left({\left(b_m\right)}_Q-b_m\right)f_2\right)\left(x_0\right)\right|\text{d}x}\\ \le \frac{1}{{\left|Q\right|}^{1+m\beta /n}}{\displaystyle {\int }_Q{E}_1\left(x\right)\text{d}x}+\frac{1}{{\left|Q\right|}^{1+m\beta /n}}{\displaystyle {\int }_Q{E}_2\left(x\right)\text{d}x}+\frac{1}{{\left|Q\right|}^{1+m\beta /n}}{\displaystyle {\int }_Q{E}_3\left(x\right)\text{d}x}+\frac{1}{{\left|Q\right|}^{1+m\beta /n}}{\displaystyle {\int }_Q{E}_4\left(x\right)\text{d}x}\\ :=N_1+N_2+N_3+N_4.\end{array}$

首先估计 $N_1$，根据引理1，可以得到

$\begin{array}{c}{N}_1\le \frac{1}{{\left|Q\right|}^{1+m\beta /n}}\underset{x\in Q}{\sup }\left|b_1\left(x\right)-{\left(b_1\right)}_Q\right|\cdots \left|b_m\left(x\right)-{\left(b_m\right)}_Q\right|{\displaystyle {\int }_Q\left|T\left(f\right)\left(x\right)\right|\text{d}x}\\ \le C{\Vert b\Vert }_{{\mathrm{Lip}}_{\beta }\left({ℝ}^n\right)}\frac{1}{{\left|Q\right|}^{1+m\beta /n}}{\left|Q\right|}^{m\beta /n}{\displaystyle {\int }_Q\left|T\left(f\right)\left(x\right)\right|\text{d}x}\\ \le C{\Vert b\Vert }_{{\mathrm{Lip}}_{\beta }\left({ℝ}^n\right)}M\left(T\left(f\right)\right)\left(x\right).\end{array}$

对于 $N_2$，选取 $1<r<p$ 并令 $\mu$，${\mu }^{\prime }$ 为整数，使得 $\mu +{\mu }^{\prime }=m$，$0\le \mu <m$，$0<{\mu }^{\prime }\le m$。根据Hölder不等式，次线性算子T在 $L^r\left({ℝ}^n\right)$ 上的有界性以及引理1，可以得到

$\begin{array}{c}{N}_2\le {\displaystyle \underset{j=1}{\overset{m-1}{\sum }}{\displaystyle \underset{\sigma \in C_j^m}{\sum }\frac{1}{{\left|Q\right|}^{1+m\beta /n}}}}{\displaystyle {\int }_Q\left|{\left(b\left(x\right)-b_Q\right)}_{\sigma }\right|}\left|T\left({\left(b-b_Q\right)}_{{\sigma }^c}f\right)\left(x\right)\right|\text{d}x\\ \le {\displaystyle \underset{j=1}{\overset{m-1}{\sum }}{\displaystyle \underset{\sigma \in C_j^m}{\sum }\frac{1}{{\left|Q\right|}^{1+m\beta /n}}{\left({\displaystyle {\int }_Q{\left|{\left(b\left(x\right)-b_Q\right)}_{\sigma }\right|}^{r/\left(r-1\right)}\text{d}x}\right)}^{1-1/r}}}{\left({\displaystyle {\int }_Q{\left|T\left({\left(b-b_Q\right)}_{{\sigma }^c}f\right)\left(x\right)\right|}^r\text{d}x}\right)}^{1/r}\end{array}$

$\begin{array}{c}\le C{\displaystyle \underset{j=1}{\overset{m-1}{\sum }}{\displaystyle \underset{\sigma \in C_j^m}{\sum }\frac{1}{{\left|Q\right|}^{1+m\beta /n}}}}{\left({\displaystyle {\int }_Q{\left|{\left(b\left(x\right)-b_Q\right)}_{\sigma }\right|}^{r/\left(r-1\right)}\text{d}x}\right)}^{1-1/r}{\left({\displaystyle {\int }_Q{\left|{\left(b\left(x\right)-b_Q\right)}_{{\sigma }^c}f\left(x\right)\right|}^r\text{d}x}\right)}^{1/r}\\ \le C{\displaystyle \underset{j=1}{\overset{m-1}{\sum }}{\displaystyle \underset{\sigma \in C_j^m}{\sum }\frac{1}{{\left|Q\right|}^{1+m\beta /n}}}}{\left|Q\right|}^{\mu \beta /n}{\Vert b_{\sigma }\Vert }_{{\mathrm{Lip}}_{\beta }({ℝ}^n)}{\left|Q\right|}^{1-1/r}{\left|Q\right|}^{{\mu }^{\prime }\beta /n}{\Vert b_{{\sigma }^c}\Vert }_{{\mathrm{Lip}}_{\beta }\left({ℝ}^n\right)}{\left|Q\right|}^{1/r}{\left(\frac{1}{\left|Q\right|}{\displaystyle {\int }_Q{\left|f\left(x\right)\right|}^r\text{d}x}\right)}^{1/r}\\ \le C{\Vert b\Vert }_{{\mathrm{Lip}}_{\beta }\left({ℝ}^n\right)}{\left(M\left({\left|f\right|}^r\right)\right)}^{1/r}\left(x\right).\end{array}$

接下来，继续估计 $N_3$。 $N_3$ 的估计与 $N_2$ 的估计类似，通过Hölder不等式，有

$\begin{array}{c}{N}_3\le C\frac{1}{{\left|Q\right|}^{1+m\beta /n}}{\left({\displaystyle {\int }_{2Q}{\left|T\left({\displaystyle \underset{j=1}{\overset{m}{\prod }}\left(b_j-{\left(b_j\right)}_Q\right)f_1}\right)\left(x\right)\right|}^r\text{d}x}\right)}^{1/r}{\left|Q\right|}^{1-1/r}\\ \le C\frac{1}{{\left|Q\right|}^{1+m\beta /n}}{\left|Q\right|}^{1-1/r}{\left({\displaystyle {\int }_{2Q}{\left|{\displaystyle \underset{j=1}{\overset{m}{\prod }}\left(\left(b_j-{\left(b_j\right)}_Q\right)f\right)\left(x\right)}\right|}^r\text{d}x}\right)}^{1/r}\\ \le C\frac{1}{{\left|Q\right|}^{1+m\beta /n}}{\left|Q\right|}^{1-1/r}{\Vert b\Vert }_{{\mathrm{Lip}}_{\beta }\left({ℝ}^n\right)}{\left|Q\right|}^{m\beta /n}{\left({\displaystyle {\int }_{2Q}{\left|f\left(x\right)\right|}^r\text{d}x}\right)}^{1/r}\\ \le C{\Vert b\Vert }_{{\mathrm{Lip}}_{\beta }\left({ℝ}^n\right)}{\left(M\left({\left|f\right|}^r\right)\right)}^{1/r}\left(x\right).\end{array}$

对于 $N_4$，因为 $y\in {\left(2Q\right)}^c$，$\left|x_0-y\right|\approx \left|x-y\right|$。根据引理1，引理2，可以得到

$\begin{array}{c}{E}_4\left(x\right)\le {\displaystyle {\int }_{{\left(2Q\right)}^c}\left|F_t\left(x,y\right)-F_t\left(x_0,y\right)\right|\left|f\left(y\right)\right|\left|{\displaystyle \underset{j=1}{\overset{m}{\prod }}\left(b_j\left(y\right)-{\left(b_j\right)}_Q\right)}\right|\text{d}y}\\ \le C{\displaystyle {\int }_{{\left(2Q\right)}^c}\frac{{\left|x_0-x\right|}^{\epsilon }}{{\left|x_0-y\right|}^{n+\epsilon }}}\left|f\left(y\right)\right|\left|{\displaystyle \underset{j=1}{\overset{m}{\prod }}\left(b_j\left(y\right)-{\left(b_j\right)}_Q\right)}\right|\text{d}y\\ \le C{\displaystyle \underset{k=1}{\overset{\infty }{\sum }}{\displaystyle {\int }_{2^{k}Q\backslash 2^{k-1}Q}{\left|x_0-x\right|}^{\epsilon }{\left|x_0-y\right|}^{-\left(n+\epsilon \right)}\left|f\left(y\right)\right|\left|{\displaystyle \underset{j=1}{\overset{m}{\prod }}\left(b_j\left(y\right)-{\left(b_j\right)}_Q\right)}\right|\text{d}y}}\\ \le C{\displaystyle \underset{k=1}{\overset{\infty }{\sum }}{2}^{-k\epsilon }{\left|2^{k}Q\right|}^{-1}}{\displaystyle {\int }_{2^{k}Q}\left|f\left(y\right)\right|}{\displaystyle \underset{j=1}{\overset{m}{\prod }}\left(\left|b_j\left(y\right)-{\left(b_j\right)}_{2^{k}Q}\right|+\left|{\left(b_j\right)}_{2^{k}Q}-{\left(b_j\right)}_Q\right|\right)\text{d}y}\\ \le C{\displaystyle \underset{k=1}{\overset{\infty }{\sum }}{2}^{-k\epsilon }{\left|2^{k}Q\right|}^{-1}{\left({\left(b_j\right)}_{2^{k}Q}-{\left(b_j\right)}_Q\right)}_{\sigma }}{\displaystyle {\int }_{2^{k}Q}\left|f\left(y\right)\right|}{\left|b_j\left(y\right)-{\left(b_j\right)}_{2^{k}Q}\right|}_{{\sigma }^c}\text{d}y\end{array}$

$\begin{array}{l}\le C{\displaystyle \underset{k=1}{\overset{\infty }{\sum }}{2}^{-k\epsilon }{\left|2^{k}Q\right|}^{-1}{\left|2^{k}Q\right|}^{\beta \mu /n}}{\Vert b_{\sigma }\Vert }_{{\mathrm{Lip}}_{\beta }\left({ℝ}^n\right)}{\left|2^{k}Q\right|}^{1+\beta {\mu }^{\prime }/n}\\ \times \frac{1}{{\left|2^{k}Q\right|}^{\beta {\mu }^{\prime }/n}}\left(\frac{1}{\left|2^{k}Q\right|}{\displaystyle {\int }_{2^{k}Q}\left|f\left(y\right)\right|}{\left|b_j\left(y\right)-{\left(b_j\right)}_{2^{k}Q}\right|}_{{\sigma }^c}\text{d}y\right)\\ \le C{\displaystyle \underset{k=1}{\overset{\infty }{\sum }}{2}^{-k\epsilon }{\left|2^{k}Q\right|}^{-1}{\left|2^{k}Q\right|}^{1+m\beta /n}{\Vert b\Vert }_{{\mathrm{Lip}}_{\beta }\left({ℝ}^n\right)}}M\left(f\right)\left(x\right)\\ \le C{\Vert b\Vert }_{{\mathrm{Lip}}_{\beta }\left({ℝ}^n\right)}{\left|Q\right|}^{m\beta /n}M\left(f\right)\left(x\right){\displaystyle \underset{k=1}{\overset{\infty }{\sum }}{2}^{\left(m\beta -\epsilon \right)k}}\\ \le C{\Vert b\Vert }_{{\mathrm{Lip}}_{\beta }\left({ℝ}^n\right)}{\left|Q\right|}^{m\beta /n}M\left(f\right)\left(x\right).\end{array}$

所以， $N_4\le C{\Vert b\Vert }_{{\mathrm{Lip}}_{\beta }\left({ℝ}^n\right)}M\left(f\right)\left(x\right)$。

结合上面对 $N_1$，$N_2$，$N_3$，$N_4$ 的估计，对所有满足 $x\in Q$ 的方体Q取上确界，可以得到

${\Vert T^b\left(f\right)\left(x\right)\Vert }_{{\stackrel{\dot{}}{K}}_{p(\cdot )}^{\alpha ,q}{\stackrel{\dot{}}{F}}_{\infty }^{m\beta }\left({ℝ}^n\right)}\le C{\Vert b\Vert }_{{\mathrm{Lip}}_{\beta }\left({ℝ}^n\right)}\left({\Vert M\left(f\right)\left(x\right)\Vert }_{{\stackrel{\dot{}}{K}}_{p(\cdot )}^{\alpha ,q}\left({ℝ}^n\right)}+{\Vert {\left(M\left({\left|f\right|}^r\right)\right)}^{1/r}\left(x\right)\Vert }_{{\stackrel{\dot{}}{K}}_{p(\cdot )}^{\alpha ,q}\left({ℝ}^n\right)}+{\Vert M\left(T\left(f\right)\left(x\right)\right)\Vert }_{{\stackrel{\dot{}}{K}}_{p(\cdot )}^{\alpha ,q}\left({ℝ}^n\right)}\right)$.

再根据引理3，引理4，可以得出

${\Vert T^b\left(f\right)\left(x\right)\Vert }_{{\stackrel{\dot{}}{K}}_{p(\cdot )}^{\alpha ,q}{\stackrel{\dot{}}{F}}_{\infty }^{m\beta }\left({ℝ}^n\right)}\le C{\Vert b\Vert }_{{\mathrm{Lip}}_{\beta }\left({ℝ}^n\right)}{\Vert f\left(x\right)\Vert }_{{\stackrel{\dot{}}{K}}_{p(\cdot )}^{\alpha ,q}\left({ℝ}^n\right)}$.

对于 $T^b$ 在非齐次变指标Herz Triebel-Lizorkin空间上有界性的证明与上述方法相似，故此处略去证明。

定理1至此估计完毕，接下来继续证明定理2。

定理2的证明：定理2的证明与定理1类似，所以令 $Q=Q\left(x_0,l\right)$，$x_0\in Q$，$l>0$。对 $f\in L^{p_1(\cdot )}\left({ℝ}^n\right)$ 做如下分解： $f_1=f{\chi }_{2Q}$，$f_2=f-f_1$。有：

$\begin{array}{l}\left|T_{\lambda }^b\left(f\right)\left(x\right)-T_{\lambda }\left(\left({\left(b_1\right)}_Q-b_1\right)\cdots \left({\left(b_m\right)}_Q-b_m\right)f_2\right)\left(x_0\right)\right|\\ \le \Vert E_t^b\left(f\right)\left(x\right)-E_t\left(\left({\left(b_1\right)}_Q-b_1\right)\cdots \left({\left(b_m\right)}_Q-b_m\right)f_2\right)\left(x_0\right)\Vert \\ \le \Vert \left(b_1\left(x\right)-{\left(b_1\right)}_Q\right)\cdots \left(b_m\left(x\right)-{\left(b_m\right)}_Q\right)E_t\left(f\right)\left(x\right)\Vert \\ +{\displaystyle \underset{j=1}{\overset{m-1}{\sum }}{\displaystyle \underset{\sigma \in C_j^m}{\sum }\Vert {\left(b\left(x\right)-b_Q\right)}_{\sigma }{E}_t\left({\left(b-b_Q\right)}_{{\sigma }^c}f\right)\left(x\right)\Vert }}\end{array}$