$L^p$ 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.

This note has been withdrawn since the main estimates are direct consequences of the restriction estimates by N. Burq, P. Gerard, and N. Tzvetkov (Duke Math. J. 2007) and L^2 concentration estimates by N. Burq and C. Zuily (Comm. Math. Phys., to appear)