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arXiv · 2609.29047

A sharp regularity threshold for Schrödinger maximal estimates on standard tori

Xi Cen·Xitao Gao·Junyong Zhang
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Abstract

We disprove almost everywhere convergence of the Schrödinger evolution on the standard torus \(\mathbb{T}^d\) for initial data \(f\in H^s(\mathbb{T}^d)\) when \(s d/(d+2)$ with $d\geq2$. 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.

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Xi Cen, Xitao Gao, Junyong Zhang. 2026-09-24. A sharp regularity threshold for Schrödinger maximal estimates on standard tori. https://arxiv.org/abs/2609.29047

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