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

 

A note on maximal averages in the plane


Authors: José A. Barrionuevo and Lucas S. Oliveira
Journal: Proc. Amer. Math. Soc. 138 (2010), 309-313
MSC (2000): Primary 42B25
Published electronically: September 4, 2009
MathSciNet review: 2550196
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Abstract: Let $ \mathcal{B}_{\delta}$ be the class of all $ h\times \delta h$ rectangles in the plane with $ h > 0$ and $ 0 < \delta < \frac{1}{2}$. The orientation of the rectangles is arbitrary. Form the maximal operator

$\displaystyle GM f(x) = \sup_{0 < \delta < \frac{1}{2}}\;\; \sup_{x\in R\in\ma... ... \frac{1}{\vert\log \delta \vert\cdot \vert R \vert}\int_R \vert f(y)\vert dy. $

Note the logarithmic term in the average. It is shown that $ GM$ is a bounded maximal operator in $ L^2(\mathbb{R}^2)$. The case of a fixed $ \delta$ is due to Córdoba.


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Additional Information

José A. Barrionuevo
Affiliation: Departamento de Matemática, Universidade Federal Rio Grande do Sul, Av. Bento Gonçalves 9500, 91509-900 Porto Alegre, RS, Brasil
Email: josea@mat.ufrgs.br

Lucas S. Oliveira
Affiliation: Departamento de Matemática, Universidade Federal Rio Grande do Sul, Av. Bento Gonçalves 9500, 91509-900 Porto Alegre, RS, Brasil
Email: lucas_gnomo@hotmail.com

DOI: http://dx.doi.org/10.1090/S0002-9939-09-10082-5
PII: S 0002-9939(09)10082-5
Keywords: Maximal operators
Received by editor(s): March 18, 2009
Received by editor(s) in revised form: April 20, 2009, and June 15, 2009
Published electronically: September 4, 2009
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.