Trace and Observability Inequalities for Laplace Eigenfunctions on the Torus

TL;DR

This paper characterizes measures on the torus for which Laplace eigenfunctions satisfy uniform trace and observability inequalities, extending classical results and applying advanced harmonic analysis tools.

Contribution

It provides necessary and sufficient conditions on measures for trace and observability inequalities of Laplace eigenfunctions on the torus, generalizing classical theorems to higher dimensions.

Findings

Characterization of measures satisfying inequalities

Extension of classical theorems to higher dimensions

Applications to quantum limits and control theory

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.