Inner Product of Eigenfunctions over Curves and Generalized Periods for Compact Riemannian Surfaces

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 $\gamma$, $|\int e_\lambda \overline{e_\mu }\,\text {d}s|$, 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 $\int e_\lambda \overline{e_\mu }\,\text {d}s=O(\mu ^\frac{1}{4})$, 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.