Improved generalized periods estimates over curves on Riemannian surfaces with nonpositive curvature

We show that, on compact Riemannian surfaces of nonpositive curvature, the generalized periods, i.e. the ν-th order Fourier coefficients of eigenfunctions eλe over a closed smooth curve ð ¾ which satisfies a natural curvature condition, go to 0 at the rate of O {equation presented} in the high energy limit λ→∞ if 0<|ν|/λ<1-δ for any fixed 0<δ<1. Our result implies, for instance, that the generalized periods over geodesic circles on any surfaces with nonpositive curvature would converge to zero at the rate of O {equation presented}.