Sharp Spectral-Cluster Restriction Bounds for Orthonormal Systems

TL;DR

This work extends sharp $L^p$ restriction bounds for eigenfunctions to systems of $L^2(M)$-orthonormal functions on a submanifold $\Sigma$ by leveraging Schatten-class duality and a wave-equation parametrix. Using approximate spectral projections and a dyadic oscillatory-integral analysis of the half-wave kernel, the authors establish essentially optimal bounds for all codimensions $\operatorname{codim}\Sigma\ge2$ and obtain refined, codimension-1 results in low and high dimensions, including logarithmic losses in borderline cases. The paper also provides sharpness results in both two and higher dimensions, showing that the derived exponents are tight in many regimes via constructions with concentrated or highly-structured orthonormal systems. Overall, the results subsume and extend Frank–Sabin’s $L^2(M)$-orthonormal bounds to spectral-cluster restrictions, highlighting the role of codimension and dimension in the optimality landscape of restriction inequalities for orthonormal systems.

Abstract

For a smooth $k$-dimensional submanifold $Σ$ of a $d$-dimensional compact Riemannian manifold $M$, we extend the $L^p(Σ)$ 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 $p\ge2$, 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.

Figures (1)

• Figure 1: $\delta(1,2,p)$ and $\alpha(1,2,p)$.

Theorems & Definitions (21)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Corollary 1.5
• Remark 1.6
• Lemma 2.1: Theorem 2.7 simon2005
• Lemma 2.2: Duality principle
• Lemma 3.1
• Remark 3.2