Pointwise bounds for $L^2$ eigenfunctions on locally symmetric spaces

TL;DR

This paper establishes pointwise bounds for $L^2$ eigenfunctions on certain locally symmetric spaces and explores implications for the $L^p$ spectrum, advancing understanding of spectral properties in geometric analysis.

Contribution

It provides new pointwise bounds for eigenfunctions on locally symmetric spaces with $ ext{Q}$-rank one, linking eigenvalue bounds to spectral properties.

Findings

Derived explicit pointwise bounds for eigenfunctions

Connected eigenfunction bounds to $L^p$ spectral results

Enhanced understanding of spectral behavior on symmetric spaces

Abstract

We prove pointwise bounds for $L^2$ eigenfunctions of the Laplace-Beltrami operator on locally symmetric spaces with $\mathbb{Q}$-rank one if the corresponding eigenvalues lie below the continuous part of the $L^2$ spectrum. Furthermore, we use these bounds in order to obtain some results concerning the $L^p$ spectrum.