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arXiv · 2610.32641

Dimension-free bounds for maximal operators associated to log kernel and applications

Mingming Cao·Honghai Liu·Zengyan Si·Haibin Yan
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Abstract

Let $M_ψ$ be the maximal operator associated to $ψ(x)=-\ln(1-|x|^2)$. We obtain dimension-free bounds on $L^p(\mathbb R^d)$ for $M_ψ$, $1<p\le\infty$. 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 $L\log L$ 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.

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Mingming Cao, Honghai Liu, Zengyan Si, Haibin Yan. 2026-10-23. Dimension-free bounds for maximal operators associated to log kernel and applications. https://arxiv.org/abs/2610.32641

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