Periodic critical-threshold estimate and convergence

Establish whether the periodic Schrödinger maximal estimate and almost-everywhere convergence for every datum in H^s(𝕋^d) hold at the critical regularity s=d/(d+2) on the standard torus 𝕋^d for d≥2.

Background

The paper proves failure of the weak L² maximal estimate and constructs divergent data for s<d/(d+2), while previously known results establish the maximal estimate and almost-everywhere convergence for s>d/(d+2). The equality case is not settled by the paper, leaving the critical periodic threshold unresolved.