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

Lack of natural weighted estimates for some singular integral operators

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

Affiliation:	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 , 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 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 weights.

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

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