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

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.