Sharp Spectral-Cluster Restriction Bounds for Orthonormal Systems

TL;DR

This paper extends restriction bounds for eigenfunctions on submanifolds to orthonormal systems, providing essentially optimal bounds across various dimensions and exponents, inspired by Frank and Sabin's work.

Contribution

It generalizes restriction bounds from individual eigenfunctions to orthonormal systems on submanifolds, achieving near-optimal results for a wide range of parameters.

Findings

Bounds are essentially optimal for most parameter ranges.

Extension from single eigenfunctions to orthonormal systems.

Inspired by and analogous to Frank and Sabin's bounds.

Abstract

For a smooth $k$-dimensional submanifold $\Sigma$ of a $d$-dimensional compact Riemannian manifold $M$, we extend the $L^p(\Sigma)$ restriction bounds of Burq-G\'erard-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 $p\ge2$, except possibly when $ d\ge3,\quad k=d-1,\quad 2\le p\le4. $ This work is inspired by a work of Frank and Sabin, who established analogous $L^p(M)$ bounds for $L^2(M)$-orthonormal systems.