On approximate differentiability of the maximal function
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- by Piotr Hajłasz and Jan Malý PDF
- Proc. Amer. Math. Soc. 138 (2010), 165-174 Request permission
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.References
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Additional Information
- Piotr Hajłasz
- Affiliation: Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, Pennsylvania 15260
- MR Author ID: 332316
- Email: hajlasz@pitt.edu
- Jan Malý
- 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
- 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 - Communicated by: Tatiana Toro
- © 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), 165-174
- MSC (2000): Primary 46E35
- DOI: https://doi.org/10.1090/S0002-9939-09-09971-7
- MathSciNet review: 2550181
Dedicated: Dedicated to Professor Bogdan Bojarski