AI 中文总结
本文证明了高维环面上薛定谔极大函数的尖锐下界，结合已有充分性结果，完全解决了周期薛定谔方程Carleson问题，并反驳了Miao-Yuan-Zhao猜想，方法基于阿贝尔簇上的复乘法。
AI 中文摘要
我们证明了在环面T^d上，对于所有维数d至少为2的薛定谔极大函数的尖锐下界，并作为推论，获得了周期薛定谔方程逐点收敛的尖锐正则性条件。结合Compaan-Lucá-Staffilani建立的充分性结果，这完全解决了所有维数至少为2的周期薛定谔方程的Carleson问题直至端点，并反驳了Miao-Yuan-Zhao的猜想正则性条件，这与R^d中的相应问题形成对比。我们还证明了高维环面方程发散集维数的尖锐估计。我们的方法使用阿贝尔簇上的复乘法。
英文摘要
We prove sharp lower bounds for the Schrödinger maximal function on the torus T^d for all dimensions d at least 2, and as a corollary obtain sharp regularity conditions for pointwise convergence of the periodic Schrödinger equation. Combined with sufficiency results established by Compaan-Lucá-Staffilani, this yields a full resolution of Carleson's problem up to endpoint for the periodic Schrödinger equation in all dimensions at least 2, and disproves the conjectured regularity condition of Miao-Yuan-Zhao, in contrast to the corresponding question in R^d. We also prove sharp estimates on the dimensions of divergence sets for the equation for high dimensional tori. Our approach uses complex multiplication on abelian varieties.
