标题： 迹和可观测性不等式在环面上的拉普拉斯本征函数 显示英文标题

标题： Trace and Observability Inequalities for Laplace Eigenfunctions on the Torus

摘要： 我们研究在d维环面上的Laplace本征函数的迹和可观测性不等式，针对任意Borel测度$\mu$。 具体而言，我们表征了使得对于Laplacian的所有本征函数$u$，不等式$$ \int |u|^2 d \mu \lesssim \int |u|^2 d x \quad \text{(trace)}, \qquad \int |u|^2 d \mu \gtrsim \int |u|^2 d x \quad \text{(observability)}$$均一致成立的测度$\mu$。 充分条件基于$\mu$的可积性和正则性推导得出，而必要条件则以测度支撑集的维度来表述。 这些结果将Zygmund和Bourgain--Rudnick的经典定理推广到了高维情形。 应用包括类似Cantor--Lebesgue定理的结果、对量子极限的约束以及Schrödinger方程的控制理论。 我们的方法结合了多种工具：球面上格点的簇结构；解耦估计；以及构造出表现出强集中或消失行为的本征函数，分别针对迹和可观测性不等式进行定制。 显示英文摘要

摘要： We investigate trace and observability inequalities for Laplace eigenfunctions on the d-dimensional torus, with respect to arbitrary Borel measures $\mu$. Specifically, we characterize the measures $\mu$ for which the inequalities $$ \int |u|^2 d \mu \lesssim \int |u|^2 d x \quad \text{(trace)}, \qquad \int |u|^2 d \mu \gtrsim \int |u|^2 d x \quad \text{(observability)}$$ hold uniformly for all eigenfunctions $u$ of the Laplacian. Sufficient conditions are derived based on the integrability and regularity of $\mu$, while necessary conditions are formulated in terms of the dimension of the support of the measure. These results generalize classical theorems of Zygmund and Bourgain--Rudnick to higher dimensions. Applications include results in the spirit of Cantor--Lebesgue theorems, constraints on quantum limits, and control theory for the Schr\"odinger equation. Our approach combines several tools: the cluster structure of lattice points on spheres; decoupling estimates; and the construction of eigenfunctions exhibiting strong concentration or vanishing behavior, tailored respectively to the trace and observability inequalities.