$L^p$ Norms of Eigenfunctions on Regular Graphs and on the Sphere

TL;DR

This paper establishes upper bounds on the $L^p$ norms of eigenfunctions for the discrete Laplacian on regular graphs and extends these results to joint eigenfunctions on the sphere, showing they match bounds known for negatively curved surfaces.

Contribution

It introduces new upper bounds for eigenfunctions on regular graphs and applies these to joint eigenfunctions on the sphere, linking discrete and continuous spectral problems.

Findings

Eigenfunctions on regular graphs have bounded $L^p$ norms.

Joint eigenfunctions on the sphere satisfy bounds similar to negatively curved surfaces.

Results connect spectral properties of graphs with geometric analysis on the sphere.

Abstract

We prove upper bounds on the $L^p$ norms of eigenfunctions of the discrete Laplacian on regular graphs. We then apply these ideas to study the $L^p$ norms of joint eigenfunctions of the Laplacian and an averaging operator over a finite collection of algebraic rotations of the $2$-sphere. Under mild conditions, such joint eigenfunctions are shown to satisfy for large $p$ the same bounds as those known for Laplace eigenfunctions on a surface of non-positive curvature.