Lack of natural weighted estimates for some singular integral operators期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

按检索

Lack of natural weighted estimates for some singular integral operators

Authors:	José Marí a Martell Carlos Pé rez Rodrigo Trujillo-Gonzá lez

Institution:	Departamento de Matemáticas, C-XV, Universidad Autónoma de Madrid, 28049 Madrid, Spain ; Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, 41080 Sevilla, Spain ; Departamento de Análisis Matemático, Universidad de La Laguna, 38271 La Laguna - S/C de Tenerife, Spain

Abstract:	We show that the classical Hörmander condition, or analogously the -Hörmander condition, for singular integral operators is not sufficient to derive Coifman's inequality $\begin{displaymath}\int_{\mathbb{R} ^n} \vert Tf(x)\vert^p\, w(x)\, dx \le C\,\int_{\mathbb{R} ^n} M f(x)^p\, w(x)\,dx, \end{displaymath}$ where $0<p<\infty$ , is the Hardy-Littlewood maximal operator, is any $A_{\infty}$ weight and is a constant depending upon and the $A_{\infty}$ constant of . This estimate is well known to hold when is a Calderón-Zygmund operator. As a consequence we deduce that the following estimate does not hold: $\begin{displaymath}\int_{\mathbb{R} ^n} \vert Tf(x)\vert^p\, w(x)\, dx \le C\,\int_{\mathbb{R} ^n} Mf(x)^p\, Mw(x)\,dx, \end{displaymath}$ where $0<p\le 1$ and where is an arbitrary weight. However, by a recent result due to A. Lerner, this inequality is satisfied whenever is a Calderón-Zygmund operator. One of the main ingredients of the proof is a very general extrapolation theorem for $A_\infty$ weights.

Keywords:	Calder\'on-Zygmund singular integral operators Muckenhoupt weights maximal functions

	点击此处可从《Transactions of the American Mathematical Society》浏览原始摘要信息
	点击此处可从《Transactions of the American Mathematical Society》下载免费的PDF全文

设为首页 | 免责声明 | 关于勤云 | 加入收藏