The L(log L)e endpoint estimate for maximal singular integral operators
We prove in this paper the following estimate for the maximal operator T ∗ associated to the singular integral operator T: kT ∗ fkL 1,∞ (w) . 1 ǫ Z Rn | f(x)| ML(log L) ǫ (w)(x) dx, w ≥ 0, 0 < ǫ ≤ 1. This follows from the sharp L p estimate kT ∗ fkLp (w) . p ′ ( 1 δ ) 1/p ′ kfk L p (ML(log L) p−1...
| Autores: | , |
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| Tipo de documento: | artigo |
| Estado: | Versión enviada para evaluación y publicación |
| Data de publicação: | 2015 |
| País: | España |
| Recursos: | Universidad de Sevilla (US) |
| Repositório: | idUS. Depósito de Investigación de la Universidad de Sevilla |
| OAI Identifier: | oai:idus.us.es:11441/42999 |
| Acesso em linha: | http://hdl.handle.net/11441/42999 https://doi.org/10.1016/j.jmaa.2015.03.017 |
| Access Level: | Acceso aberto |
| Palavra-chave: | maximal operators Calderón–Zygmund operators weighted estimates |
| Resumo: | We prove in this paper the following estimate for the maximal operator T ∗ associated to the singular integral operator T: kT ∗ fkL 1,∞ (w) . 1 ǫ Z Rn | f(x)| ML(log L) ǫ (w)(x) dx, w ≥ 0, 0 < ǫ ≤ 1. This follows from the sharp L p estimate kT ∗ fkLp (w) . p ′ ( 1 δ ) 1/p ′ kfk L p (ML(log L) p−1+δ (w)), 1 < p < ∞, w ≥ 0, 0 < δ ≤ 1. As as a consequence we deduce that kT ∗ fkL 1,∞ (w) . [w]A1 log(e + [w]A∞ ) Z Rn | f | w dx, extending the endpoint results obtained in [LOP] A. Lerner, S. Ombrosi and C. Pérez, A1 bounds for Calderón-Zygmund operators related to a problem of Muckenhoupt and Wheeden, Mathematical Research Letters (2009), 16, 149–156 and [HP] T. Hytönen and C. Pérez, Sharp weighted bounds involving A∞, Analysis and P.D.E. 6 (2013), 777–818. DOI 10.2140/apde.2013.6.777 to maximal singular integrals. Another consequence is a quantitative two weight bump estimate. |
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