Weighted geodesic restrictions of arithmetic eigenfunctions

Abstract

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