$L^p$ norms of higher rank eigenfunctions and bounds for spherical functions

TL;DR

This paper establishes near-optimal bounds on the $L^p$ norms of eigenfunctions on compact symmetric spaces, using advanced harmonic analysis and new asymptotic bounds for spherical functions.

Contribution

It introduces novel asymptotic bounds for spherical functions and combines semiclassical analysis with harmonic theory to derive sharp eigenfunction norm estimates.

Findings

Almost sharp $L^p$ bounds for eigenfunctions

New asymptotic bounds for spherical functions

Bounds for eigenfunction restrictions to flat subspaces

Abstract

We prove almost sharp upper bounds for the $L^p$ norms of eigenfunctions of the full ring of invariant differential operators on a compact locally symmetric space, as well as their restrictions to maximal flat subspaces. Our proof combines techniques from semiclassical analysis with harmonic theory on reductive groups, and makes use of new asymptotic bounds for spherical functions that are of independent interest.