On the Central Maximal Function in ℝn
Abstract
We consider the central Hardy--Littlewood maximal operator in ℝn. For every integer n≥1, we construct two bounded, nonnegative functions f,g∈ L1(ℝn) that differ on a set of positive measure and satisfy Mcf(x)=Mcg(x) at every point x∈ℝn. Thus the central maximal operator is not injective on the nonnegative cone of L1(ℝn). The functions can have compact support; when n≥2, they can also be chosen radial, either with compact support or strictly positive throughout the space. The construction makes a small redistribution of mass invisible to the maximal operator by excluding averaging spheres that cross its support. We also give an explicit formula, in terms of the incomplete beta function, for the n-dimensional volume of the intersection of two balls with arbitrary radii and centers, and an elementary formula for the central maximal function of a ball in ℝ3.
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BibTeX
@misc{solyanik2026central,
title = {{On the Central Maximal Function in $\mathbbR^n$}},
author = {Alexey Solyanik},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.27687},
eprint = {2609.27687},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27687},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the Central Maximal Function in ℝ^n
AU - Alexey Solyanik
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2609.27687
UR - https://arxiv.org/abs/2609.27687
AB - We consider the central Hardy--Littlewood maximal operator in ℝ^n. For every integer n≥1, we construct two bounded, nonnegative functions f,g∈ L^1(ℝ^n) that differ on a set of positive measure and satisfy M_cf(x)=M_cg(x) at every point x∈ℝ^n. Thus the central maximal operator is not injective on the nonnegative cone of L^1(ℝ^n). The functions can have compact support; when n≥2, they can also be chosen radial, either with compact support or strictly positive throughout the space. The construction makes a small redistribution of mass invisible to the maximal operator by excluding averaging spheres that cross its support. We also give an explicit formula, in terms of the incomplete beta function, for the n-dimensional volume of the intersection of two balls with arbitrary radii and centers, and an elementary formula for the central maximal function of a ball in ℝ^3.
ER -