On the growth of generalized Fourier coefficients of restricted eigenfunctions

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