Inner product of eigenfunctions over curves and generalized periods for compact Riemannian surfaces

TL;DR

This paper establishes sharp bounds on the inner products of eigenfunctions over curves on compact Riemannian surfaces, unifying period and restriction estimates, and extends results to higher dimensions and Fourier coefficients.

Contribution

It provides new sharp bounds for eigenfunction inner products over curves, generalizes existing results, and extends the analysis to higher dimensions and Fourier coefficients.

Findings

Inner product bounds are sharp on spheres and tori.

Unified estimates for period integrals and restriction problems.

Extension of results to higher-dimensional hypersurfaces.

Abstract

We show that for a smooth closed curve $\gamma$ on a compact Riemannian surface without boundary, the inner product of two eigenfunctions $e_\lambda$ and $e_\mu$ restricted to $\gamma$, $|\int e_\lambda\overline{e_\mu}\,ds|$, is bounded by $\min\{\lambda^\frac12,\mu^\frac12\}$. Furthermore, given $0<c<1$, if $0<\mu<c\lambda$, we prove that $\int e_\lambda\overline{e_\mu}\,ds=O(\mu^\frac14)$, which is sharp on the sphere $S^2$. These bounds unify the period integral estimates and the $L^2$-restriction estimates in an explicit way. Using a similar argument, we also show that the $\nu$-th order Fourier coefficient of $e_\lambda$ over $\gamma$ is uniformly bounded if $0<\nu<c\lambda$, which generalizes a result of Reznikov for compact hyperbolic surfaces, and is sharp on both $S^2$ and the flat torus $\mathbb T^2$. Moreover, we show that the analogs of our results also hold in higher dimensions for the inner product of eigenfunctions over hypersurfaces.