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arXiv · 2504.13019

The higher regularity of the discrete Hardy-Littlewood maximal function

Faruk Temur·Hikmet Burak Özcan
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Abstract

In a recent short note the first author gave the first positive result on the higher order regularity of the discrete noncentered Hardy-Littlewood maximal function. In this article we conduct a thorough investigation of possible similar results for higher order derivatives. We uncover that such results are indeed a consequence of a stronger phenomenon regarding the growth of $l^p(\Z)$ norms of the derivatives of characteristic functions of finite subsets of $\Z$. Along the way we discover very interesting connections to Prouhot-Tarry-Escott (PTE) problem, and to zeros of complex polynomials with restricted coefficients (Littlewood-type polynomials).

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Faruk Temur, Hikmet Burak Özcan. 2025-04-17. The higher regularity of the discrete Hardy-Littlewood maximal function. https://arxiv.org/abs/2504.13019

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