\ell^p boundedness for the discrete spherical maximal operator over odd radii

Determine whether the discrete spherical maximal operator on \(\mathbb Z^d\), with the supremum restricted to odd squared radii \(n\), is bounded on \(\ell^p(\mathbb Z^d)\) for the relevant exponents and dimensions.

Background

For dimensions $d\geq5$, the unrestricted discrete spherical maximal operator is known to be bounded on $\ell^p$ exactly when $p>d/(d-2)$. In dimension four, irregularities in the number of representations as sums of four squares cause the unrestricted maximal operator to be unbounded for every finite $p$.

The paper identifies the odd-radius restriction in this dimension-four setting as a major unresolved problem. The quoted passage does not assert a resolution of the odd-radius problem.