The paper proves that on the d-torus, trace and observability inequalities for Laplace eigenfunctions require the measure's support to have dimension at least d-2, and gives Fourier-decay and Sobolev-regularity conditions under which they hold.
Restriction of toral eigenfunctions to totally geodesic submanifolds
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Trace and Observability Inequalities for Laplace Eigenfunctions on the Torus
The paper proves that on the d-torus, trace and observability inequalities for Laplace eigenfunctions require the measure's support to have dimension at least d-2, and gives Fourier-decay and Sobolev-regularity conditions under which they hold.