PreprintCC BY-SA 4.0arXiv
Sharp derivative inequalities for maximal functions
Abstract
We determine the optimal constant in the Lp-derivative inequality for the uncentered Hardy--Littlewood maximal operator on the real line for every 1<p<∞. It is the unique number γp∈ [0,1] satisfying (2p−1)γp+(p−1)γp(2p−1)/(p−1)=p. As p goes from $1$ to +∞, γp decreases from $1$ to √2−1, and at the endpoints the limiting arguments recover the classical total-variation and Lipschitz inequalities, with the qualitative regularity ingredient at the BV endpoint supplied by Aldaz and Pérez Lázaro.
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BibTeX
@misc{liao2026sharp,
title = {{Sharp derivative inequalities for maximal functions}},
author = {Sung-Yi Liao and José Madrid and Eyvindur Ari Palsson},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39178},
eprint = {2610.39178},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39178},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Sharp derivative inequalities for maximal functions
AU - Sung-Yi Liao
AU - José Madrid
AU - Eyvindur Ari Palsson
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39178
UR - https://arxiv.org/abs/2610.39178
AB - We determine the optimal constant in the L^p-derivative inequality for the uncentered Hardy--Littlewood maximal operator on the real line for every 1<p<∞. It is the unique number γ_p∈ [0,1] satisfying (2p−1)γ_p+(p−1)γ_p^(2p−1)/(p−1)=p. As p goes from $1$ to +∞, γ_p decreases from $1$ to √2−1, and at the endpoints the limiting arguments recover the classical total-variation and Lipschitz inequalities, with the qualitative regularity ingredient at the BV endpoint supplied by Aldaz and Pérez Lázaro.
ER -