摘要
综述回顾了带有非倍测度的欧氏空间R^d上的Calderon-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∈(0,d]以及C使得对任意的x∈R^d和r∈(0,∞),μ(B(x,r))≤Cr^n.回顾的主要结果包括:Hardy空间H^1(μ)与正则BMO空间RBMO(μ);与H^1(μ)以及RBMO(μ)相关的插值定理;Calderon-Zygmund分解;T(1)定理与Calderon-Zygmund算子在Lebesgue空间和Hardy空间上的有界性;Cotlar不等式与极大Calderon-Zygmund算子的有界性;多线性Calderon-Zygmund算子在乘积Lebesgue空间上的性质;Calderon-Zygmund算子的加权模不等式;由Calderon-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.
In this survey, the authors review some basic results on Calderon-Zygmund theory on R^d with the non-doubling Radon measure #. Instead of the usual doubling condition, the measure μ is only assumed to satisfy some growth condition, that is, there exist positive con- stants n ∈ （0, d] and C such that p（B（x, r）） 〈 Crn for all x ∈ R^d and r E （0, ∞）. 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 H^1（μ） and RBMO（μ）, the Calderon- Zygmund decomposition, the T（1） theorem and the properties of the Calderon-Zygmund operator on Lebesgue spaces and Hardy spaces, Cotlar＇s inequality and the boundedness of the maximal Calderon-Zygmund operator, the behaviors of the multilinear Calderon-Zygmund operator on Lebesgue spaces, weighted norm inequalities for the CalderSn-Zygmund operator, the boundedness of the commutator generated by the Calderon-Zygmund operator and RBMO（μ） function. Moreover, some new progress obtained recently is also included.
出处
《数学进展》
CSCD
北大核心
2013年第4期417-440,共24页
Advances in Mathematics（China）
基金
The first author is supported by NSFC(No.10971228)and the second author supported by NSFC(No.11171027)
作者简介
E-mail： Corresponding author： mengyan@ruc.edu.cn
