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On the regularity of maximal operators

Authors: Emanuel Carneiro and Diego Moreira
Journal: Proc. Amer. Math. Soc. 136 (2008), 4395-4404
MSC (2000): Primary 42B25, 54C08, 46E35
Published electronically: July 28, 2008
MathSciNet review: 2431055
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Abstract: We study the regularity of the bilinear maximal operator when applied to Sobolev functions, proving that it maps $ W^{1,p}(\mathbb{R}) \times W^{1,q}(\mathbb{R}) \to W^{1,r}(\mathbb{R})$ with $ 1 <p,q < \infty$ and $ r\geq 1$, boundedly and continuously. The same result holds on $ \mathbb{R}^n$ when $ r>1$. We also investigate the almost everywhere and weak convergence under the action of the classical Hardy-Littlewood maximal operator, both in its global and local versions.

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Additional Information

Emanuel Carneiro
Affiliation: Department of Mathematics, University of Texas at Austin, Austin, Texas 78712-1082.

Diego Moreira
Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242

Keywords: Maximal operator, bilinear maximal, Sobolev spaces, weak differentiability, weak continuity
Received by editor(s): November 20, 2007
Published electronically: July 28, 2008
Additional Notes: The first author was supported by CAPES/FULBRIGHT grant BEX 1710-04-4.
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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