A note on maximal averages in the plane
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- by José A. Barrionuevo and Lucas S. Oliveira PDF
- Proc. Amer. Math. Soc. 138 (2010), 309-313 Request permission
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 \[ GM f(x) = \sup _{0 < \delta < \frac {1}{2}}\;\; \sup _{x\in R\in \mathcal {B}_{\delta }}\;\; \frac {1}{|\log \delta |\cdot | R |}\int _R |f(y)| dy. \] 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.References
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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
- 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
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 309-313
- MSC (2000): Primary 42B25
- DOI: https://doi.org/10.1090/S0002-9939-09-10082-5
- MathSciNet review: 2550196