On the growth of generalized Fourier coefficients of restricted eigenfunctions

Abstract

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

25 pages, 2 figures