L2 to Lp bounds for spectral projectors on thin intervals in Riemannian manifolds

TL;DR

This paper reviews known results on L2 to Lp bounds for spectral projectors on thin intervals in Riemannian manifolds, emphasizing harmonic analytic methods and the influence of manifold geometry.

Contribution

It provides a focused review of existing results, especially in symmetric cases, highlighting harmonic analysis approaches over microlocal techniques.

Findings

Global geometry affects bounds for smaller windows

Connections to differential geometry, combinatorics, and number theory

Emphasis on harmonic analytic methods

Abstract

Given a Riemannian manifold endowed with its Laplace-Beltrami operator, consider the associated spectral projector on a thin interval. As an operator from L2 to Lp, what is its operator norm? For a window of size 1, this question is fully answered by a celebrated theorem of Sogge, which applies to any manifold. For smaller windows, the global geometry of the manifold comes into play, and connections to a number of mathematical fields (such as Differential Geometry, Combinatorics, Number Theory) appear, but the problem remains mostly open. The aim of this article is to review known results, focusing on cases exhibiting symmetry and emphasizing harmonic analytic rather than microlocal methods.