PreprintCC BY 4.0arXiv
On the variation and asymptotics of maximal functions
Abstract
In this paper we present partial progress towards two conjectures on the variation of maximal operators of convolution type Mφ, associated to non-negative and radial decreasing kernels φ of integral $1$. In dimension d=1 we prove that var(Mφ f) ≤ var(f) when f is a characteristic function. In higher dimensions we prove that var(Mφ f) ≤ Cd var(f) when f is a radial characteristic function, with a constant Cd depending only on the dimension d but not on the kernel φ. We further show that $C_d \sim \sqrt{d}\log d $ is the sharp dependence on d for such an inequality. This last claim relies on a novel explicit asymptotic estimate for the Hardy-Littlewood maximal function of the unit ball in ℝd, a result of independent interest.
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BibTeX
@misc{carneiro2026variation,
title = {{On the variation and asymptotics of maximal functions}},
author = {Emanuel Carneiro and Julian Weigt},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18544},
eprint = {2610.18544},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18544},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the variation and asymptotics of maximal functions
AU - Emanuel Carneiro
AU - Julian Weigt
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18544
UR - https://arxiv.org/abs/2610.18544
AB - In this paper we present partial progress towards two conjectures on the variation of maximal operators of convolution type M_φ, associated to non-negative and radial decreasing kernels φ of integral $1$. In dimension d=1 we prove that var(M_φ f) ≤ var(f) when f is a characteristic function. In higher dimensions we prove that var(M_φ f) ≤ C_d var(f) when f is a radial characteristic function, with a constant C_d depending only on the dimension d but not on the kernel φ. We further show that $C_d \sim \sqrt{d}\log d $ is the sharp dependence on d for such an inequality. This last claim relies on a novel explicit asymptotic estimate for the Hardy-Littlewood maximal function of the unit ball in ℝ^d, a result of independent interest.
ER -