Corpus ID: 52553123

# Improved Generalized Periods estimates on Riemannian Surfaces with Nonpositive Curvature

@article{Xi2017ImprovedGP,
title={Improved Generalized Periods estimates on Riemannian Surfaces with Nonpositive Curvature},
author={Yakun Xi},
journal={arXiv: Analysis of PDEs},
year={2017}
}
• Yakun Xi
• Published 27 November 2017
• Mathematics
• arXiv: Analysis of PDEs
We show that on compact Riemann surfaces of negative curvature, the generalized periods, i.e. the $\nu$-th order Fourier coefficient of eigenfunctions $e_\lambda$ over a period geodesic $\gamma$ goes to 0 at the rate of $O((\log\lambda)^{-1/2})$, if $0<\nu<c_0\lambda$, given any $0<c_0<1$. No such result is possible for the sphere $S^2$ or the flat torus $\mathbb T^2$. Our proof consists of a further refinement of a recent paper by Sogge, Xi and Zhang on the geodesic period integrals (\$\nu=0… Expand
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