A sharp regularity threshold for Schrödinger maximal estimates on standard tori
Abstract
We disprove almost everywhere convergence of the Schrödinger evolution on the standard torus 𝕋d for initial data f∈ Hs(𝕋d) when s<d/(d+2), in all dimensions d≥2. We construct normalized data with frequencies of size N whose evolution attains size Nd/(d+2) on a set of uniformly positive measure. The key ingredient in the construction is that the frequencies lie in an affine congruence class, so that at suitable rational times their phases coincide on a family of well-separated spatial points. The result matches the known estimate for s>d/(d+2) with d≥2. By integer dilation and uniform boundedness, we also obtain a single datum in Hs whose evolution is unbounded along a sequence of times tending to zero whenever s<d/(d+2). We also record a logarithmic upper bound in 2D at the critical frequency power and a lower bound on shrinking time intervals.
Related
- Orthonormal maximal estimates for fractional Schrödinger operators along Hölder curves
- Maximal estimates of the Schrödinger operators with quadratic potential for orthonormal systems
- Schrödinger estimates with arbitrarily small periodic regularity on ℍ2 × 𝕋
- Maximal regularity and caloric trace estimates in mixed Lebesgue norms for the heat equation
- Lp-regularity for fourth-order Schrödinger operators with unbounded or degenerate coefficients
BibTeX
@misc{cen2026sharp,
title = {{A sharp regularity threshold for Schrödinger maximal estimates on standard tori}},
author = {Xi Cen and Xitao Gao and Junyong Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29047},
eprint = {2609.29047},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29047},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A sharp regularity threshold for Schrödinger maximal estimates on standard tori
AU - Xi Cen
AU - Xitao Gao
AU - Junyong Zhang
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29047
UR - https://arxiv.org/abs/2609.29047
AB - We disprove almost everywhere convergence of the Schrödinger evolution on the standard torus 𝕋^d for initial data f∈ H^s(𝕋^d) when s<d/(d+2), in all dimensions d≥2. We construct normalized data with frequencies of size N whose evolution attains size N^d/(d+2) on a set of uniformly positive measure. The key ingredient in the construction is that the frequencies lie in an affine congruence class, so that at suitable rational times their phases coincide on a family of well-separated spatial points. The result matches the known estimate for s>d/(d+2) with d≥2. By integer dilation and uniform boundedness, we also obtain a single datum in H^s whose evolution is unbounded along a sequence of times tending to zero whenever s<d/(d+2). We also record a logarithmic upper bound in 2D at the critical frequency power and a lower bound on shrinking time intervals.
ER -