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

 

On approximate differentiability of the maximal function


Authors: Piotr Hajłasz and Jan Maly
Journal: Proc. Amer. Math. Soc. 138 (2010), 165-174
MSC (2000): Primary 42B25; Secondary 46E35, 31B05
Published electronically: September 3, 2009
MathSciNet review: 2550181
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Abstract: We prove that if $ f\in L^1(\mathbb{R}^n)$ is approximately differentiable a.e., then the Hardy-Littlewood maximal function $ \mathcal{M}f$ is also approximately differentiable a.e. Moreover, if we only assume that $ f\in L^1(\mathbb{R}^n)$, then any open set of $ \mathbb{R}^n$ contains a subset of positive measure such that $ \mathcal{M} f$ is approximately differentiable on that set. On the other hand we present an example of $ f\in L^1(\mathbb{R})$ such that $ \mathcal{M}f$ is not approximately differentiable a.e.


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Additional Information

Piotr Hajłasz
Affiliation: Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, Pennsylvania 15260
Email: hajlasz@pitt.edu

Jan Maly
Affiliation: Department KMA of the Faculty of Mathematics and Physics, Charles University, Sokolovská 83, CZ-18675 Praha 8, Czech Republic – and – Department of Mathematics of the Faculty of Science, J. E. Purkyně University, České mládeže 8, 400 96 Ústí nad Labem, Czech Republic
Email: maly@karlin.mff.cuni.cz

DOI: http://dx.doi.org/10.1090/S0002-9939-09-09971-7
PII: S 0002-9939(09)09971-7
Received by editor(s): February 18, 2009
Published electronically: September 3, 2009
Additional Notes: The first author was supported by NSF grant DMS-0500966.
The second author was supported by the research project MSM 0021620839 and by grants GA ČR 201/06/0198, 201/09/0067
Dedicated: Dedicated to Professor Bogdan Bojarski
Communicated by: Tatiana Toro
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.