Almost everywhere convergence for Lebesgue differentiation processes along rectangles

TL;DR

This paper investigates the almost everywhere convergence of Lebesgue differentiation processes along shrinking rectangles in the plane, revealing cases where convergence fails in certain function spaces and connecting to directional maximal operator properties.

Contribution

It introduces new examples of differentiation processes that fail to converge a.e. in $L^ olinebreak ext{infinity}$, highlighting the relationship with directional maximal operators.

Findings

Examples of non-converging differentiation processes in $L^ olinebreak ext{infinity}$.

Failure of boundedness of directional maximal operators for certain slopes.

Counterpart to known boundedness results for directional maximal operators.

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 $1\leq p<\infty$.