On the growth of generalized Fourier coefficients of restricted eigenfunctions

Abstract

Let (M, g) be a smooth, compact, Riemannian manifold and {φ h } a sequence of L 2 -normalized Laplace eigenfunctions on M . For a smooth submanifold H ⊂ M , we consider the growth of the restricted eigenfunctions φ h | H by testing them against a sequence of functions {ψ h } on H whose wavefront set avoids S * H. That is, we study what we call the generalized Fourier coefficients: φ h , ψ h L 2 (H) . We give an explicit bound on these coefficients depending on how the defect measures for the two collections of functions φ h and ψ h relate. This allows us to get a little−o improvement whenever the collection of recurrent directions over the wavefront set of ψ h is small. To obtain our estimates, we utilize geodesic beam techniques.