The higher regularity of the discrete Hardy-Littlewood maximal function

2025/04/17 by Faruk Temur, Temur, Faruk, Hikmet Burak Özcan +1
Mathematics · #Holomorphic and Operator Theory #Advanced Banach Space Theory #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.2504.13019

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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