Improved Generalized Periods Estimates Over Curves on Riemannian Surfaces with Nonpositive Curvature

We show that on compact Riemann surfaces of nonpositive curvature, the generalized periods, i.e. the $\nu$-th order Fourier coefficients of eigenfunctions $e_{\lambda }$ over a closed smooth curve $\gamma$ which satisfies a natural curvature condition, go to 0 at the rate of $O((\log \lambda {)}^{-1\mathrm{/}2})$, if $0<\mathrm{\mid }\nu \mathrm{\mid }\mathrm{/}\lambda <1-\delta$, for any fixed $0<\delta <1$. Our result implies, for instance, the generalized periods over geodesic circles on any surfaces with nonpositive curvature would converge to zero at the rate of $O((\log \lambda {)}^{-1\mathrm{/}2})$. A direct corollary of our results and the QER theorem of Toth and Zelditch is that for a geodesic circle $\gamma$ on a compact hyperbolic surface, the restriction $e_{{\lambda }_j}{\mathrm{\mid }}_{\gamma }$ of an orthonormal basis $\{e_{{\lambda }_j}\}$ has a full density subsequence that goes to zero in weak-$L^2(\gamma )$. One key step of our proof is a microlocal decomposition of the measure over $\gamma$ into tangential and transversal parts.