Trace and Observability Inequalities for Laplace Eigenfunctions on the Torus

Abstract

We investigate trace and observability inequalities for Laplace eigenfunctions on the d-dimensional torus, with respect to arbitrary Borel measures $\hat{I}¼$. Specifically, we characterize the measures $\hat{I}¼$ for which the inequalities $$\int |u{|}^{2}d\hat{I}¼\lesssim \int |u{|}^{2}dx\text{(trace)},\int |u{|}^{2}d\hat{I}¼\gtrsim \int |u{|}^{2}dx\text{(observability)}$$ hold uniformly for all eigenfunctions $u$ of the Laplacian. Sufficient conditions are derived based on the integrability and regularity of $\hat{I}¼$, 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 Schrödinger 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.

22 pages