analysis

Sharp higher order regularity of discrete maximal functions

arXiv:2607.10753

summary

The paper obtains sharp ℓ^p(Z) bounds for the first and second discrete derivatives of the uncentered Hardy‑Littlewood maximal operator applied to binary functions on the integers, and provides lower bounds for higher‑order derivatives.

Abstract

We derive sharp $ℓ^{p} (Z)$ bounds for the $k$ th derivative of the discrete uncentered maximal operator applied to characteristic functions $f : Z \to {0, 1}$ in the cases $k = 0, 1, 2$ . When $k = 1, 2$ these are the first sharp bounds for derivatives of a Hardy-Littlewood maximal function in continuous or discrete settings when $1 < p < \infty$ . We also establish several lower bounds for $k \geq 3$ and for general functions $f : Z \to R$ .

22 pages, 2 tables

Topics & keywords

#discrete maximal functions #hardy-littlewood maximal operator #ℓ^p bounds #higher order derivatives #harmonic analysisℓ^p(Z)uncentered maximal operatorcharacteristic functionssharp boundskth derivative