The second derivative of the discrete Hardy-Littlewood maximal function
Faruk Temur
TL;DR
This work delivers the first positive higher-order regularity result for the discrete noncentered Hardy-Littlewood maximal function by bounding the second discrete derivative of $M\\chi_A$ in terms of the second discrete derivative of $\\chi_A$. It develops a discrete calculus with the second derivative, convex/concave boundary structure, and a boundary-driven estimate to prove the central $L^p$ bound $\\| (M\\chi_A)'' \\|_p \\le 2^{1-1/p} 3^{1/p} \\|\\chi_A''\\|_p$ for $1 \\le p \\le \\infty$. A pivotal lemma asserts that concavity of $M\\chi_A$ can occur only at points of $A$, enabling a sharp $p=1$ bound and the full range of $p$ through the argument. Overall, the paper advances discrete regularity theory for maximal operators and informs the variation properties of discrete maximal functions.
Abstract
The regularity of the Hardy-Littlewood maximal function, in both discrete and continuous contexts, and for both centered and noncentered variants, has been subjected to intense study for the last two decades. But efforts so far have concentrated on first order differentiability and variation, as it is known that in the continuous context higher order regularity is impossible. This short note gives the first positive result on the higher order regularity of the discrete noncentered maximal function.