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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Weighted $ L^p$ boundedness of Carleson type maximal operators


Authors: Yong Ding and Honghai Liu
Journal: Proc. Amer. Math. Soc. 140 (2012), 2739-2751
MSC (2010): Primary 42B20, 42B25; Secondary 42B99
Published electronically: December 8, 2011
MathSciNet review: 2910762
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Abstract: In 2001, E. M. Stein and S. Wainger gave the $ L^p$ boundedness of the Carleson type maximal operator $ \mathcal {T}^\ast $, which is defined by

$\displaystyle \mathcal {T}^\ast f(x)=\sup _\lambda \bigg \vert\int _{{\mathbb{R}}^n}e^{iP_\lambda (y)}K(y)f(x-y)dy\bigg \vert.$

In this paper, the authors show that if $ K$ is a homogeneous kernel, i.e. $ K(y)=\Omega (y')\vert y\vert^{-n}$, then Stein-Wainger's result still holds on the weighted $ L^p$ spaces when $ \Omega $ satisfies only an $ L^q$-Dini condition for some $ 1<q\le \infty $.

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Additional Information

Yong Ding
Affiliation: School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems (BNU), Beijing Normal University, Ministry of Education of China, Beijing 100875, People’s Republic of China
Email: dingy@bnu.edu.cn

Honghai Liu
Affiliation: School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, People’s Republic of China
Email: hhliu@hpu.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11110-9
PII: S 0002-9939(2011)11110-9
Keywords: Carleson operator, homogeneous kernel, $L^{q}$-Dini condition, $A_{p}$ weight
Received by editor(s): May 29, 2010
Received by editor(s) in revised form: March 6, 2011
Published electronically: December 8, 2011
Additional Notes: The first author was supported by the NSF of China (Grant 10931001), SRFDP of China (Grant 20090003110018) and Program for Changjiang Scholars and Innovative Research Team in University.
Communicated by: Richard Rochberg
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.