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International Mathematics Research Notices (2008) Vol. 2008 : article ID rnm161, 11 pages, doi:10.1093/imrn/rnm161 published on January 15, 2008
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© The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org

Sharp A1 Bounds for Calderón-Zygmund Operators and the Relationship with a Problem of Muckenhoupt and Wheeden

Andrei K. Lerner1, Sheldy Ombrosi2 and Carlos Pérez3

1 Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, 41080 Sevilla, Spain
2 Departamento de Matemática Universidad Nacional del Sur, Bahía Blanca, 8000, Argentina
3 Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, 41080 Sevilla, Spain

Correspondence: Correspondence to be sent to: carlosperez{at}us.es

For any Calderón–Zygmund operator T the following sharp estimate is obtained for 1 < p < {infty}:


Formula

where Formula . In the case where p = 2 and T is a classical convolution singular integral, this result is due to R. Fefferman and J. Pipher [7]. Then, we deduce the following weak type (1, 1) estimate related to a problem of Muckenhoupt and Wheeden [11]:


Formula

where w isin A1 and {varphi}(t) = t(1 + log+ t)(1 + log+ log+ t).

Communicated by Prof. Carlos Kenig


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