On the growth of generalized Fourier coefficients of restricted eigenfunctions

TL;DR

This paper investigates how the Fourier coefficients of eigenfunctions restricted to submanifolds grow, providing bounds based on defect measures and using geodesic beam techniques for improved estimates.

Contribution

It introduces explicit bounds on generalized Fourier coefficients of restricted eigenfunctions, relating them to defect measures and recurrent directions, with novel application of geodesic beam methods.

Findings

Derived bounds depend on defect measure relations.

Achieved small-o improvements with limited recurrent directions.

Utilized geodesic beam techniques for estimates.

Abstract

Let $(M,g)$ be a smooth, compact, Riemannian manifold and $\{\phi_h\}$ 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 $\phi_h|_H$ by testing them against a sequence of functions $\{\psi_h\}$ on $H$ whose wavefront set avoids $S^*H$. That is, we study what we call the generalized Fourier coefficients: $\langle \phi_h,\psi_h\rangle_{L^2(H)}$. We give an explicit bound on these coefficients depending on how the defect measures for the two collections of functions $\phi_h$ 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.