$L^p$ concentration estimates for the Laplacian eigenfunctions near submanifolds

TL;DR

This paper establishes sharp $L^p$ bounds for Laplacian eigenfunctions near submanifolds on compact Riemannian manifolds, revealing how neighborhood size affects spectral projection estimates and extending previous results.

Contribution

It provides new sharp $L^p$ estimates for spectral projections near submanifolds, especially for neighborhood sizes proportional to $ ext{frequency}^{- ext{power}}$, generalizing prior work.

Findings

Sharp estimates for $ ext{neighborhood size} \, \mathcal{O}( ext{frequency}^{- ext{delta}})$ with $ ext{delta} \ge 1$

Optimality of Sogge's estimates for $0 \le \delta \le 1/2$

Partial sharpness results for intermediate $ ext{delta}$ values between 1/2 and 1

Abstract

We study $L^p$ bounds on spectral projections for the Laplace operator on compact Riemannian manifolds, restricted to small frequency dependent neighborhoods of submanifolds. In particular, if $\lambda$ is a frequency and the size of the neigborhood is $\mathcal{O}(\lambda^{-\delta})$, then new sharp estimates are established when $\delta\ge 1$, while for $0\le \delta\le 1/2$, Sogge's estimates turn out to be optimal. In the intermediate region $1/2<\delta<1$, we sometimes get sharp estimates as well. Our arguments follow closely a recent work by Burq and Zuily.