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E.M. Stein’s dimension-free maximal-function question for discrete Euclidean balls

For every 1<p≤∞1<p\leq\infty, does there exist a constant Cp<∞C_p<\infty, independent of the dimension dd, such that for every f∈ℓp(Zd)f\in\ell^p(\mathbb{Z}^d), ∥sup⁡t≥0∣Mtf∣∥ℓp(Zd)≤Cp∥f∥ℓp(Zd)\left\|\sup_{t\geq 0}\left|M_t f\right|\right\|_{\ell^p(\mathbb{Z}^d)}\leq C_p\|f\|_{\ell^p(\mathbb{Z}^d)}, where Mtf(x)=∣Bt∩Zd∣−1∑y∈Bt∩Zdf(x−y)M_t f(x)=\lvert B_t\cap\mathbb{Z}^d\rvert^{-1}\sum_{y\in B_t\cap\mathbb{Z}^d}f(x-y) and Bt={y∈Rd:∣y∣2≤t}B_t=\{y\in\mathbb{R}^d:\lvert y\rvert_2\leq t\}?

References

Primary source

arXiv

Additional references

discrete-analysisergodic-theoryharmonic-analysis
Posted · edited

Progress summary

Refreshed
Claimed solved

A September 2026 preprint claims to settle Stein’s question, but no independent verification was found.

Stein asked whether averaging over every discrete Euclidean ball can be bounded by one constant that works in every dimension. Earlier work established only restricted-radius or dyadic variants, not the full question.

Known results

  • Dyadic radii, all dimensions and p∈[2,∞]p\in[2,\infty] (Kosz, Mirek, and collaborators, 2018).
  • Dyadic discrete spherical averages, d≥5d\ge 5 and p∈[2,∞]p\in[2,\infty] (2020).
  • Small scales t≤d1/2−εt\le d^{1/2-\varepsilon}, with dimension-free bounds for p∈[2,∞]p\in[2,\infty] (2025).
  • Further small-scale and dyadic results for ℓq\ell^q balls, q≥2q\ge 2 (2024).

September 9, 2026 claimed resolution

A submission titled Dimension-free estimates for the full discrete Euclidean ball maximal function claims a dimension-independent bound for the full-radius operator, which would resolve Stein’s question. The claim is unverified: no independent confirmation or peer review was retrieved.

Current status (as of September 2026): The full-radius question is claimed solved by a September 2026 preprint, but the result remains unverified; earlier partial results do not settle it.

Sources

Solutions 0

No solutions have been posted yet.