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.