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Lack of natural weighted estimates for some singular integral operators

Authors: José María Martell, Carlos Pérez and Rodrigo Trujillo-González
Journal: Trans. Amer. Math. Soc. 357 (2005), 385-396
MSC (2000): Primary 42B20, 42B25
Published electronically: August 11, 2004
MathSciNet review: 2098100
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Abstract: We show that the classical Hörmander condition, or analogously the $L^r$-Hörmander condition, for singular integral operators $T$ 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 $0<p<\infty$, $M$ is the Hardy-Littlewood maximal operator, $w$ is any $A_{\infty}$ weight and $C$ is a constant depending upon $p$ and the $A_{\infty}$ constant of $w$. This estimate is well known to hold when $T$ 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 $w$ is an arbitrary weight. However, by a recent result due to A. Lerner, this inequality is satisfied whenever $T$ is a Calderón-Zygmund operator.

One of the main ingredients of the proof is a very general extrapolation theorem for $A_\infty$ weights.

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Additional Information

José María Martell
Affiliation: Departamento de Matemáticas, C-XV, Universidad Autónoma de Madrid, 28049 Madrid, Spain

Carlos Pérez
Affiliation: Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, 41080 Sevilla, Spain

Rodrigo Trujillo-González
Affiliation: Departamento de Análisis Matemático, Universidad de La Laguna, 38271 La Laguna - S/C de Tenerife, Spain

Keywords: Calder\'on-Zygmund singular integral operators, Muckenhoupt weights, maximal functions
Received by editor(s): May 23, 2003
Received by editor(s) in revised form: September 18, 2003
Published electronically: August 11, 2004
Additional Notes: The first author was partially supported by MCYT Grant BFM2001-0189
The second author was partially supported by DGICYT Grant PB980106
The third author was supported by MCYT Grant BFM2002-02098
Article copyright: © Copyright 2004 American Mathematical Society

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