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

Abstract

We show that for a smooth closed curve $γ$ on a compact Riemannian surface without boundary, the inner product of two eigenfunctions $e_λ$ and $e_μ$ restricted to $γ$, $|\int e_λ\overline{e_μ}\,ds|$, is bounded by $\min\{λ^\frac12,μ^\frac12\}$. Furthermore, given $0<c<1$, if $0<μ<cλ$, we prove that $\int e_λ\overline{e_μ}\,ds=O(μ^\frac14)$, 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 $ν$-th order Fourier coefficient of $e_λ$ over $γ$ is uniformly bounded if $0<ν<cλ$, 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.