Dimension-free full maximal inequality for 1-symmetric convex bodies

Prove that the full discrete maximal operators associated with all 1-symmetric convex bodies satisfy a dimension-free ell^p bound for every pin[2,cinfty], namely that the supremum over dimensions, 1-symmetric convex bodies, and dilation parameters is uniformly bounded.

Background

The paper establishes dimension-free estimates for dyadic maximal operators over 1-symmetric convex bodies and for the full maximal operator in the small-scale regime. It does not establish the corresponding estimate for the full supremum over all positive dilation parameters.

The conjecture asks whether these partial results extend to the complete discrete maximal function, uniformly in the ambient dimension and in the choice of 1-symmetric convex body.

References

We do not address here whether the dimension-free estimate in Theorem \ref{thm:main:sphere} extends below p=2.

— Dimension-free estimates for full discrete maximal functions associated with Euclidean balls and spheres  (2609.10763 - Hormozi et al., 9 Sep 2026) in Section 1, Statement of the results

Due to these partial results he conjectured an analogue of Stein's question for 1-symmetric convex bodies. A resolution of this conjecture would give a discrete analogue of Bourgain's celebrated result .

— Dimension-free estimates for full discrete maximal functions associated with Euclidean balls and spheres  (2609.10763 - Hormozi et al., 9 Sep 2026) in Section 1, A brief history, subsection Discrete Euclidean balls and Theorem~\ref{thm:main}

From the perspective of Theorems \ref{dyadic dim-free}, \ref{small scale dim-free} it seems natural to conjecture the following. We have

\sup_{p \in [2,\infty]} \sup_{d \in N} \sup_{G \in \mathcal{F}d} \Big| \sup{t>0} |\mathcal{M}{t}G| \Big|{\ellp(Zd) \to \ellp(Zd)}< \infty.

— High-dimensional discrete 1-symmetric convex bodies and dimension-free estimates for maximal functions  (2608.17302 - Niksiński, 18 Aug 2026) in Section 1, subsection 1.4, Conjecture 1.4 (labelled “dim free conjecture”)

In the mid 1990s Stein asked the following question. Is it true that

\sup_{d \in N} \Big| \sup_{t>0} |\mathcal{M}{t}{B{2,(d)}| \Big|{\ell2(Zd) \to \ell2(Zd)}< \infty ?

— High-dimensional discrete 1-symmetric convex bodies and dimension-free estimates for maximal functions  (2608.17302 - Niksiński, 18 Aug 2026) in Section 1, subsection 1.4, Question 1.5

We finish this part with the following question. Is it true that for all $K>0$ there exists $d \in N$ and $G \subseteq Rd$ a closed symmetric convex body, which is permutation invariant such that

\Big| \sup_{t>0} |\mathcal{M}tG| \Big|{\ell2(Zd) \to \ell2(Zd)} \ge K.

We conjecture, although we currently have no supporting evidence, that the question above has affirmative answer.

— High-dimensional discrete 1-symmetric convex bodies and dimension-free estimates for maximal functions  (2608.17302 - Niksiński, 18 Aug 2026) in Section 1, subsection 1.4, final Question