Weighted geodesic restrictions of arithmetic eigenfunctions

2025/12/02 by Hou, Jiaqi, Huang, Xiaoqi
#11F03 #58J50 #Analysis of PDEs (math.AP) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2512.03291

Abstract

Let X be an arithmetic hyperbolic surface, ψ a Hecke-Maass form, ℓ a geodesic segment on X, and μ a Borel measure supported on ℓ with dimension greater than 1/2. We obtain a power saving over the local bound of Eswarathasan and Pramanik for the L² norm of ψ with respect to μ, which is a weighted generalization of Marshall's geodesic restriction bound and is proved by applying the method of arithmetic amplification. On a general 2-dimensional Riemannian manifold, we also obtain a Kakeya-Nikodym bound for the L² norm of any Laplace-Beltrami eigenfunction with respect to a Borel measure supported on a geodesic segment with dimension greater than 1/2.

Citations

• Refined L^p restriction estimate for eigenfunctions on Riemannian surfaces
• Quasimode concentration on compact space forms
• Curvature and sharp growth rates of log-quasimodes on compact manifolds
• On logarithmic improvements of critical geodesic restriction bounds in the presence of nonpositive curvature
• Concerning Toponogov's Theorem and logarithmic improvement of estimates\n of eigenfunctions
• A few endpoint geodesic restriction estimates for eigenfunctions
• Restriction of toral eigenfunctions to hypersurfaces and nodal sets
• Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds

Related

• The Geodesic Restriction Problem for Arithmetic Spherical Harmonics
• Restrictions of Maass forms on SL(2,ℂ) to hyperbolic surfaces and geodesic tubes
• Norms of geodesic restrictions for eigenfunctions on hyperbolic surfaces and representation theory
• A few endpoint geodesic restriction estimates for eigenfunctions
• Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds