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Weighted Endpoint Estimates for Commutators ofCalder'on-Zygmund Operators
LIANG YIYU 1, KY LUONG DANG 2, YANG DACHUN 3
1. Department of Mathematics,Beijing Jiaotong University,Beijing 100044, People's Republic of China
2. Department of Mathematics,University of Quy Nhon,170 An Duong Vuong,Quy Nhon, Binh Dinh, Vietnam
3. School of Mathematical Sciences, Beijing Normal University,Laboratory of Mathematics and Complex Systems, Ministry ofEducation, Beijing 100875, People's Republic of China
*Correspondence author
#Submitted by
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Funding: Luong Dang Ky is supported by Vietnam National Foundation for Science and Technology Development (Grant No. 101.02-2014.31).Da（No.Grant Nos. 11571039 and 11361020）, the Specialized Research Fund for the Doctoral Programof Higher Education of China (Grant No. 20120003110003) andthe Fundamental ResearchF（No.Grant Nos. 2014KJJCA10）
Opened online:12 October 2015
Accepted by: none
Citation: LIANG YIYU, KY LUONG DANG, YANG DACHUN.Weighted Endpoint Estimates for Commutators ofCalder'on-Zygmund Operators[OL]. [12 October 2015] http://en.paper.edu.cn/en_releasepaper/content/4656709
 
 
Let $deltain(0,1]$ and $T$ be a $delta$-Calder'on-Zygmund operator.Let $w$ be in the Muckenhoupt class $A_{1+delta/n}({mathbb R}^n)$ satisfying$int_{{mathbb R}^n} rac {w(x)}{1+|x|^n},dx<infty$.When $bin{ m BMO}(mathbb R^n)$,it is well known that the commutator $[b, T]$ is not bounded from $H^1(mathbb R^n)$to $L^1(mathbb R^n)$ if $b$ is not a constant function.In this article, the authors find out a proper subspace${mathopmathcal{BMO}_w({mathbb R}^n)}$of $mathopmathrm{BMO}(mathbb R^n)$ such that,if $bin {mathopmathcal{BMO}_w({mathbb R}^n)}$, then $[b,T]$ is bounded from theweighted Hardy space $H_w^1(mathbb R^n)$ to the weighted Lebesguespace $L_w^1(mathbb R^n)$.Conversely, if $bin{ m BMO}({mathbb R}^n)$ and the commutators of theclassical Riesz transforms ${[b,R_j]}_{j=1}^n$are bounded from $H^1_w({mathbb R}^n)$ into $L^1_w(R^n)$,then $bin {mathopmathcal{BMO}_w({mathbb R}^n)}$.
Keywords:Calder'on-Zygmund operator, commutator, Muckenhoupt weight,BMO space, Hardy space
 
 
 

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