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arXiv · 2207.02176

Almost everywhere convergence for Lebesgue differentiation processes along rectangles

Emma D'Aniello·Anthony Gauvan·Laurent Moonens·Joseph M. Rosenblatt
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Abstract

In this paper, we study Lebesgue differentiation processes along rectangles $R_k$ shrinking to the origin in the Euclidean plane, and the question of their almost everywhere convergence in $L^p$ spaces. In particular, classes of examples of such processes failing to converge a.e. in $L^\infty$ are provided, for which $R_k$ is known to be oriented along the slope $k^{-s}$ for $s>0$, yielding an interesting counterpart to the fact that the directional maximal operator associated to the set $\{k^{-s}:k\in\mathbb{N}^*\}$ fails to be bounded in $L^p$ for any $1\leq p<\infty$.

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Emma D'Aniello, Anthony Gauvan, Laurent Moonens, Joseph M. Rosenblatt. 2022-07-05. Almost everywhere convergence for Lebesgue differentiation processes along rectangles. https://arxiv.org/abs/2207.02176

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