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arXiv · 2609.08433

Full-radius dimension-free maximal inequalities for discrete Euclidean balls

Kaiwen Jin·Qingtang Su
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Abstract

For every $1<p\le\infty$, we prove dimension-free maximal inequalities over all radii for normalized averages over Euclidean balls in $\mathbb Z^d$. In particular, this settles the $\ell^2$ question attributed to Stein. The proof uses a two-saddle expansion at integer squared radii to compare ball multipliers with normalized discrete Gaussians at the zero and parity frequencies. First-order estimates give the full-radius $\ell^2$ bound. For $1<p<2$, we combine higher-order residual estimates with dimension-uniform $\ell^1$ bounds for the residuals and levelwise interpolation to obtain the full range.

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Kaiwen Jin, Qingtang Su. 2026-09-08. Full-radius dimension-free maximal inequalities for discrete Euclidean balls. https://arxiv.org/abs/2609.08433

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