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综述
非倍测度空间上的Calder\'on-Zygmund算子
Calder\'on-Zygmund Operators Typeset the title in boldface with the first letter of important words capitalized With Non-doubling Measures

胡国恩1,孟岩2
HU Guoen1,MENG Yan2,*

1. 郑州信息科学与技术研究所应用数学系, 郑州, 河南, 450002;
2. 中国人民大学信息学院, 北京, 100872
1. Department of Applied Mathematics,Zhengzhou Information Science and Technology Institute, Zhengzhou, Henan, 450002, P. R. China;
2. School of Information, Renmin University of China, Beijing, 100872, P. R. China

收稿日期: 2012-03-12
出版日期: 2013-08-25
DOI: 10.11845/sxjz.20130401a

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摘要 综述回顾了带有非倍测度的欧氏空间$\mathbb{R}^d$上的Calder\'on-Zygmund理论中的基本结果. 在该背景下欧氏空间上所赋予的测度$μ$不需要满足通常的双倍条件, 只需满足如下增长性条件, 即存在正常数$n\in(0,d]$以及$C$使得对任意的$x\in \mathbb{R}^d$和$r\in(0,\infty)$, $μ(B(x, r))\leq Cr^n$. 回顾的主要结果包括: Hardy空间H1(μ)与正则BMO空间 RBMO}(μ); 与H1(μ)以及 RBMO}(μ)相关的插值定理; Calder\'on-Zygmund分解; $T(1)$定理与 Calder\'on-Zygmund算子在Lebesgue空间和Hardy空间上的有界性; Cotlar不等式与极大Calder\'on-Zygmund算子的有界性; 多线性Calder\'on-Zygmund算子在乘积Lebesgue空间上的性质; Calder\'on-Zygmund算子的加权模不等式; 由Calder\'on-Zygmund算子与 RBMO}(μ)函数所生成的交换子的有界性. 此外,作者还介绍了该研究方面的一些最新进展与成果.
关键词 非双倍测度Calder\'on-Zygmund算子Lebesgue空间H1(μ)RBMO(μ)    
Abstract:In this survey, the authors review some basic results on Calder\'on-Zygmund theory on $\mathbb{R}^{d}$ with the non-doubling Radon measure μ. Instead of the usual doubling condition, the measure μ is only assumed to satisfy some growth condition, that is, there exist positive constants $n\in(0,\,d]$ and $C$ such that $μ(B(x,r))\le Cr^n$ for all $x\in\mathbb{R}^{d}$ and $r\in(0,\,\infty)$. In this setting, the results that we review are the following: the Hardy space H1(μ) and the regularized BMO space RBMO(μ), some interpolation results related to H1(μ) and RBMO(μ), the Calder\'on-Zygmund decomposition, the T(1) theorem and the properties of the Calder\'on-Zygmund operator on Lebesgue spaces and Hardy spaces, Cotlar's inequality and the boundedness of the maximal Calder\'on-Zygmund operator, the behaviors of the multilinear Calder\'on-Zygmund operator on Lebesgue spaces, weighted norm inequalities for the Calder\'on-Zygmund operator, the boundedness of the commutator generated by the Calder\'on-Zygmund operator and RBMO(μ) function. Moreover, some new progress obtained recently is also included.
Key wordsCalder\'on-Zygmund operator    Lebesgue space    H1(μ)    RBMO(μ)
基金资助:The first author is supported by NSFC (No. 10971228) and the second author is supported by NSFC (No. 11171027).
通讯作者: *mengyan@ruc.edu.cn   
[1] 沈聪辉, 徐景实. Banach空间值的变指标Lebesgue空间上的Calderón-Zygmund算子[J]. 数学进展, 2017, 46(3): 441-452.
[2] 卢盛栋,江寅生. 非双倍测度 Marcinkiewicz积分交换子和多线性交换子的弱型估计(英)[J]. 数学进展, 2014, 43(1): 103-117.
[3] 王松柏,江寅生,李宝德. Marcinkiewicz积分在RBMO上的有界性 (英)[J]. 数学进展, 2012, 41(4): 447-454.
[4] 王玮;徐景实;. 变指数Lebesgue积空间上的多重线性奇异积分与Lipschitz函数生成的交换子(英文)[J]. 数学进展, 2009, 38(6): 669-677.
[5] 韩彦昌;. 非齐型空间上一类积分算子在Morrey空间的有界性[J]. 数学进展, 2008, 37(5): 527-538.
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