Improved generalized periods estimates over curves on Riemannian surfaces with non positive curvature

We show that, on compact Riemannian surfaces of nonpositive curvature, the generalized periods, i.e. the v-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) (-1/2)) in the high energy limit lambda -> infinity if 0 < vertical bar v vertical bar/lambda< 1 - delta for any fixed 0 < delta < 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((log lambda)(-1/2)).