The second derivative of the discrete Hardy-Littlewood maximal function

Faruk Temur

doi:10.26650/ijmath.2025.00026

The second derivative of the discrete Hardy-Littlewood maximal function

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

References

Bober, J., Carneiro, E., Hughes, K., Pierce, L.B., 2012, On a discrete version of Tanaka’s theorem for maximal functions, Proc. Amer. Math. Soc., 140 no. 5, 1669-1680. google scholar
Carneiro, E., 2019, Regularity of maximal operators: recent progress and some open problems, in New Trends in Applied Harmonic Analysis, Volume 2: Harmonic Analysis, Geometric Measure Theory, and Applications; Editors: Aldroubi, A., Cabrelli, C., Jaffard, S., Molter, U., Springer. google scholar

Details

Primary Language

English

Subjects

Pure Mathematics (Other)

Journal Section

Research Article

Authors

Faruk Temur ^*
0000-0003-1519-4082
Türkiye

Publication Date

December 16, 2025

Submission Date

August 20, 2025

Acceptance Date

October 6, 2025

Published in Issue

Year 2025 Volume: 3 Number: 2

DOI

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

IZ

https://izlik.org/JA97WC79RT

Cite

RIS / Bibtex

APA

Temur, F. (2025). The second derivative of the discrete Hardy-Littlewood maximal function. Istanbul Journal of Mathematics, 3(2), 45-47. https://doi.org/10.26650/ijmath.2025.00026

AMA

1.Temur F. The second derivative of the discrete Hardy-Littlewood maximal function. Istanbul Journal of Mathematics. 2025;3(2):45-47. doi:10.26650/ijmath.2025.00026

Chicago

Temur, Faruk. 2025. “The Second Derivative of the Discrete Hardy-Littlewood Maximal Function”. Istanbul Journal of Mathematics 3 (2): 45-47. https://doi.org/10.26650/ijmath.2025.00026.

EndNote

Temur F (December 1, 2025) The second derivative of the discrete Hardy-Littlewood maximal function. Istanbul Journal of Mathematics 3 2 45–47.

IEEE

[1]F. Temur, “The second derivative of the discrete Hardy-Littlewood maximal function”, Istanbul Journal of Mathematics, vol. 3, no. 2, pp. 45–47, Dec. 2025, doi: 10.26650/ijmath.2025.00026.

ISNAD

Temur, Faruk. “The Second Derivative of the Discrete Hardy-Littlewood Maximal Function”. Istanbul Journal of Mathematics 3/2 (December 1, 2025): 45-47. https://doi.org/10.26650/ijmath.2025.00026.

JAMA

1.Temur F. The second derivative of the discrete Hardy-Littlewood maximal function. Istanbul Journal of Mathematics. 2025;3:45–47.

MLA

Temur, Faruk. “The Second Derivative of the Discrete Hardy-Littlewood Maximal Function”. Istanbul Journal of Mathematics, vol. 3, no. 2, Dec. 2025, pp. 45-47, doi:10.26650/ijmath.2025.00026.

Vancouver

1.Faruk Temur. The second derivative of the discrete Hardy-Littlewood maximal function. Istanbul Journal of Mathematics. 2025 Dec. 1;3(2):45-7. doi:10.26650/ijmath.2025.00026