标题： 三维环面的谱投影的界 显示英文标题

标题： Bounds for spectral projectors on the three-dimensional torus

摘要： 我们研究在频谱窗口较窄的情况下，环面上欧几里得拉普拉斯算子的频谱投影算子的$L^2$到$L^p$范数。对于固定大小的窗口，这是 Sogge 的经典结果；在小窗口极限情况下，我们只剩下拉普拉斯算子特征函数的$L^p$范数，如 Bourgain 所考虑的那样。对于三维环面，我们证明了前两位作者之前提出的关于这些范数大小的猜想的新情况；我们还改进了某些先前的结果，以消除所有维度中的$\epsilon$-损失。我们使用数论的方法：数的几何、圆法和 Guo 提出的指数和界。我们结合高度分割和双线性论证来证明精确的结果。我们阐述了所使用各种技术及其局限性。 显示英文摘要

摘要： We study $L^2$ to $L^p$ operator norms of spectral projectors for the Euclidean Laplacian on the torus in the case where the spectral window is narrow. With a window of constant size this is a classical result of Sogge; in the small-window limit we are left with $L^p$ norms of eigenfunctions of the Laplacian, as considered for instance by Bourgain. For the three-dimensional torus we prove new cases of a previous conjecture of the first two authors concerning the size of these norms; we also refine certain prior results to remove $\epsilon$-losses in all dimensions. We use methods from number theory: the geometry of numbers, the circle method and exponential sum bounds due to Guo. We complement these techniques with height splitting and a bilinear argument to prove sharp results. We exposit on the various techniques used and their limitations.