Euclidean Schrödinger maximal-estimate endpoint

Determine whether the Euclidean Schrödinger maximal estimate and the associated almost-everywhere convergence result hold at the endpoint regularity s=d/[2(d+1)] for dimensions d≥2.

Background

For the Schrödinger equation on Euclidean space, the paper summarizes matching necessary and sufficient results away from the endpoint regularity s=d/[2(d+1)]. The endpoint case is identified as unresolved for dimensions d≥2, independently of the periodic-torus problem studied in the paper.

References

The endpoint case $s=d/[2(d+1)]$ remains open for $d\ge2$.

— A sharp regularity threshold for Schrödinger maximal estimates on standard tori  (2609.29047 - Cen et al., 24 Sep 2026) in Section 1, subsection “Background and motivations”

The endpoint estimate is therefore not decided by the preceding bounds.

— A sharp regularity threshold for Schrödinger maximal estimates on standard tori  (2609.29047 - Cen et al., 24 Sep 2026) in Section 4, subsection “The 2D frequency estimate”