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The norm of the centered Hardy--Littlewood maximal operator
Abstract
We determine the exact Lp(ℝ) norm of the centered Hardy--Littlewood maximal operator for every 1<p<∞, proving that it equals p/(p−1). This settles the sharp strong-type problem for the centered maximal operator on the real line. For every p in this range, the norm is strictly larger than the constant associated with the critical power |x|−1/p, thereby disproving the conjecture of Dror, Ganguli, and Strichartz that this profile determines the unrestricted norm.
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BibTeX
@misc{madrid2026norm,
title = {{The norm of the centered Hardy--Littlewood maximal operator}},
author = {José Madrid},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.12440},
eprint = {2609.12440},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12440},
}
RIS
TY - UNPB
TI - The norm of the centered Hardy--Littlewood maximal operator
AU - José Madrid
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2609.12440
UR - https://arxiv.org/abs/2609.12440
AB - We determine the exact L^p(ℝ) norm of the centered Hardy--Littlewood maximal operator for every 1<p<∞, proving that it equals p/(p−1). This settles the sharp strong-type problem for the centered maximal operator on the real line. For every p in this range, the norm is strictly larger than the constant associated with the critical power |x|^−1/p, thereby disproving the conjecture of Dror, Ganguli, and Strichartz that this profile determines the unrestricted norm.
ER -