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

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.