Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 42B25, 46E35, 31B05

Retrieve articles in all journals with MSC (2000): 42B25, 46E35, 31B05

Additional Information

Piotr Hajłasz
Affiliation: Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, Pennsylvania 15260

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

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.

American Mathematical Society