Eigenfunction maxima and spherical means

TL;DR

This paper establishes an equivalence between Hörmander $L^2$-$L^{ obreak ext{infty}}$ estimates and restriction estimates to small geodesic spheres for certain manifolds, linking eigenfunction behavior to geometric analysis.

Contribution

It demonstrates the equivalence between two fundamental types of estimates for eigenfunctions on specific manifolds, advancing understanding in geometric analysis.

Findings

Hörmander estimates are equivalent to restriction estimates on small geodesic spheres

The results connect eigenfunction bounds with geometric restriction phenomena

Provides new tools for analyzing Laplacian eigenfunctions on manifolds

Abstract

The eigenfunctions of the Laplacian are a central object from the realms of analytic number theory to geometric analysis. We prove that H\"ormander $L^2$-$L^{\infty}$ estimates are equivalent to restriction estimates to small geodesic spheres for a certain class of manifolds.