Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

On approximate differentiability of the maximal function

Author(s): Piotr Hajłasz; Jan Maly
Journal: Proc. Amer. Math. Soc. 138 (2010), 165-174.
MSC (2000): Primary 42B25; Secondary 46E35, 31B05
Posted: 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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