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

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.

arXiv comment51 pages, no figures. This version is not intended for journal submission