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

 

Endpoint estimates for the circular maximal function


Author: Sanghyuk Lee
Journal: Proc. Amer. Math. Soc. 131 (2003), 1433-1442
MSC (2000): Primary 42B25; Secondary 35L05
Published electronically: September 19, 2002
MathSciNet review: 1949873
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Abstract: We consider the problem of endpoint estimates for the circular maximal function defined by

\begin{displaymath}Mf(x)=\sup_{1<t<2}\left\vert\int_{S^1} f(x-ty)d\sigma(y)\right\vert \end{displaymath}

where $d\sigma$ is the normalized surface area measure on $S^1$. Let $\Delta$ be the closed triangle with vertices $(0,0), (1/2, 1/2), (2/5,1/5)$. We prove that for $(1/p,1/q)\in \Delta\setminus\{(1/2,1/2), (2/5,1/5)\}$, there is a constant $C$such that $ \Vert Mf\Vert _{L^q(\mathbb R^2)}\le C\Vert f\Vert _{L^p(\mathbb R^2)}.$ Furthermore, $\Vert Mf\Vert _{L^{5,\infty}(\mathbb R^2)}\le C \Vert f\Vert _{L^{5/2,1}(\mathbb R^2)}$.


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Additional Information

Sanghyuk Lee
Affiliation: Department of Mathematics, Pohang University of Science and Technology, Pohang 790-784, Korea
Email: huk@euclid.postech.ac.kr

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06781-3
PII: S 0002-9939(02)06781-3
Keywords: Circular maximal function, endpoint estimates
Received by editor(s): June 12, 2001
Received by editor(s) in revised form: December 7, 2001
Published electronically: September 19, 2002
Additional Notes: The author was partially supported by the BK21 Project (PI: Jong-Guk Bak).
Communicated by: Andreas Seeger
Article copyright: © Copyright 2002 American Mathematical Society