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 $\gamma$, $\mathrm{\mid }\int e_{\lambda }\overline{e_{\mu }}ds\mathrm{\mid }$, is bounded by $\min \{{\lambda }^{\frac{1}{2}},{\mu }^{\frac{1}{2}}\}$. Furthermore, given $0<c<1$, if $0<\mu <c\lambda$, we prove that $\int e_\lambda\overline{e_\mu},ds=O(\mu^\frac14)$,whichissharponthesphere$S^2$\mathrm{.}Theseboundsunifytheperiodintegralestimatesandthe$L^2$-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.