Sharp spectral-cluster restriction bounds for orthonormal systems

Abstract

Let $\Sigma$ be a smooth submanifold of a compact Riemannian manifold $M$. We study the restriction to $\Sigma$ of densities generated by $L^2(M)$-orthonormal systems of eigenfunctions whose eigenfrequencies lie in a unit-width spectral cluster. For every $p\geq2$, we establish $L^{p/2}(\Sigma)$ bounds that quantify the gain arising from orthogonality and determine the optimal summability exponent for the coefficient sequence in all dimensions and codimensions. The summability exponent is fully sharp, while the frequency dependence is optimal modulo logarithmic factors. In the codimension-one case when $\dim M\geq3$, the optimal summability exponent exhibits a new, dimension-independent critical point at $p=4$, in addition to the classical critical point $p=\frac{2d}{d-1}$. This phenomenon has no analogue in the corresponding restriction estimates for a single eigenfunction. We also obtain improved, essentially sharp estimates when $\Sigma$ is a curve with nonvanishing geodesic curvature on a Riemannian surface.