The second derivative of the discrete Hardy-Littlewood maximal function

TL;DR

This paper proves for the first time that the discrete noncentered Hardy-Littlewood maximal function exhibits higher order regularity, advancing understanding of its differentiability properties.

Contribution

It provides the first positive result on higher order regularity of the discrete noncentered maximal function, filling a gap in the regularity theory.

Findings

First positive higher order regularity result for discrete noncentered maximal function

Advances understanding of differentiability in discrete harmonic analysis

Contrasts with continuous case where higher order regularity is impossible

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.