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arXiv · 2507.16599

Trace and Observability Inequalities for Laplace Eigenfunctions on the Torus

Nicolas Burq·Pierre Germain·Massimo Sorella·Hui Zhu
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Abstract

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.

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Nicolas Burq, Pierre Germain, Massimo Sorella, Hui Zhu. 2025-07-22. Trace and Observability Inequalities for Laplace Eigenfunctions on the Torus. https://arxiv.org/abs/2507.16599

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