On approximate differentiability of the maximal function

Hajłasz, Piotr; Malý, Jan

doi:10.1090/S0002-9939-09-09971-7

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.

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