The second derivative of the discrete Hardy-Littlewood maximal function

Year 2025, Volume: 3 Issue: 2, 45 - 47, 16.12.2025

Faruk Temur

https://doi.org/10.26650/ijmath.2025.00026

https://izlik.org/JA97WC79RT

Abstract

The regularity of the Hardy-Littlewood maximal function, in both discrete and continuous contexts, and for both centered and non-centered 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 non-centered maximal function. We demonstrate that for the class of characteristic functions it is possible to obtain second order regularity. The method of proof relies on the convexity properties of the maximal function, the boundary properties of subsets of integers, and a discrete analogue of the fundamental theorem of calculus.

Keywords

Boundedness of variation , discrete maximal function , second derivative

References

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