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

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