Stein’s dimension-free maximal inequality problem for discrete Euclidean balls

Stein’s problem asks whether the discrete Hardy–Littlewood maximal operator over Euclidean balls has bounds independent of the dimension, including the full supremum over radii. The newly reported claim addresses Stein’s explicitly identified $\ell^2$ question and extends it across the stated exponent range.

Known results

• Earlier work established dimension-free bounds for dyadic radii and $p\in[2,\infty]$, but not the full-radius inequality.
• Subsequent work treated small dyadic scales and related $\ell^q$ balls, without resolving the full Euclidean-ball problem.

September 2026 claimed solution

A September 2026 arXiv preprint, Full-radius dimension-free maximal inequalities for discrete Euclidean balls, claims the full-radius dimension-free inequality across the entire stated exponent range. It is presented as an unrefereed preprint, and no independent verification or reported correction was found.

Current status (as of September 2026): The full-radius problem is claimed solved by an unrefereed preprint, but the claim remains unverified.