Sharp Spectral-Cluster Restriction Bounds for Orthonormal Systems

For a smooth $k$-dimensional submanifold $\mathrm{\Sigma }$ of a $d$-dimensional compact Riemannian manifold $M$, we extend the $L^p(\mathrm{\Sigma })$ restriction bounds of Burq-Gérard-Tzvetkov – originally proved for individual Laplace–Beltrami eigenfunction – to arbitrary systems of $L^2(M)$-orthonormal functions. Our bounds are essentially optimal for every triple $(k,d,p)$ with $p\ge 2$, except possibly when $d\ge 3,k=d-1,2\le p\le 4.$ This work is inspired by a work of Frank and Sabin, who established analogous $L^p(M)$ bounds for $L^2(M)$-orthonormal systems.