d维环面上薛定谔算子扰动的极大估计
Maximal estimates for perturbations of the Schrödinger operator on $\mathbb{T}^d$
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中文总结 AI 辅助
该研究针对d维环面，证明周期薛定谔方程的猜想极大估计在抛物面小扰动下不成立，利用 incidence 估计新下界结合齐性动力学给出替代证明，且估计在抛物面解耦端点处本质最优。
中文摘要 AI 辅助
受环面（$\boldsymbol{\text{T}^d}$）上薛定谔极大估计的启发，我们研究与$\boldsymbol{C^2}$图超曲面相关的指数和的$\boldsymbol{L^p_x L^\frac{1}{\text{infinity}}_t}$极大估计。我们证明，当允许抛物面有小扰动时，周期薛定谔方程的猜想极大估计不成立，这可视为Fu、Ren和Wang所证现象的高维推广。我们的方法利用了Cairo和Zhang最初证明的 incidence 估计的新下界，为此我们基于齐性动力学提供了另一种证明。此外，这些估计在抛物面解耦端点$\boldsymbol{p = \frac{2(d+2)}{d}}$处本质上是最优的。
英文摘要
We study $L^p_x L^\infty_t$ maximal estimates for exponential sums associated to $C^2$ graph hypersurfaces, motivated by Schrödinger maximal estimates on $\mathbb{T}^d$. We show that the conjectured maximal estimate for the periodic Schrödinger equation fails when one allows small perturbations of the paraboloid, which can be viewed as a higher-dimensional extension of the phenomenon proved by Fu, Ren, and Wang. Our approach uses new lower bounds for incidence estimates originally proven by Cairo and Zhang, for which we provide an alternative proof based on homogeneous dynamics. Moreover the estimates are essentially sharp at the decoupling endpoint for the paraboloid $p = \frac{2(d+2)}{d}$.
