PreprintCC BY 4.0arXiv
Dimension-free bounds for maximal operators associated to log kernel and applications
Abstract
Let Mψ be the maximal operator associated to ψ(x)=−ln(1−|x|2). We obtain dimension-free bounds on Lp(ℝd) for Mψ, 1<p≤∞. We show that the weak L1 estimate does not hold for Mψ, but does hold for the corresponding lacunary maximal operator MψD. An weak Llog L type estimate for Mψ is obtained, which seems to be the right substitution for the weak L1 estimate. Estimates for Mψ are applied to dimension-free estimates for maximal Riesz transforms obtained by Mateu, Verdera, Orobitg and Pérez.
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BibTeX
@misc{cao2026dimensionfree,
title = {{Dimension-free bounds for maximal operators associated to log kernel and applications}},
author = {Mingming Cao and Honghai Liu and Zengyan Si and Haibin Yan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32641},
eprint = {2610.32641},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32641},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Dimension-free bounds for maximal operators associated to log kernel and applications
AU - Mingming Cao
AU - Honghai Liu
AU - Zengyan Si
AU - Haibin Yan
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32641
UR - https://arxiv.org/abs/2610.32641
AB - Let M_ψ be the maximal operator associated to ψ(x)=−ln(1−|x|^2). We obtain dimension-free bounds on L^p(ℝ^d) for M_ψ, 1<p≤∞. We show that the weak L^1 estimate does not hold for M_ψ, but does hold for the corresponding lacunary maximal operator M_ψ^D. An weak Llog L type estimate for M_ψ is obtained, which seems to be the right substitution for the weak L^1 estimate. Estimates for M_ψ are applied to dimension-free estimates for maximal Riesz transforms obtained by Mateu, Verdera, Orobitg and Pérez.
ER -