Endpoint eigenfunction restriction estimates in codimension two

Abstract

We investigate the optimal endpoint $L^2$ restriction estimates of Laplace eigenfunctions on submanifolds of codimension 2 in a smooth closed Riemannian manifold $M$. For every fixed smooth codimension-two submanifold $Σ$, we prove a little-oh improvement $o(λ^{1/2}\sqrt{\logλ})$ on the classical estimate $O(λ^{1/2}\sqrt{\logλ})$ of Burq--Gérard--Tzvetkov and Hu. Our proof uses the Gaussian transform and Taylor approximation to reduce the problem to polynomial models that can be handled by Stein--Street's estimate on singular Radon transforms. The three-dimensional case can be handled directly by Ricci--Stein's estimate. Moreover, we construct explicit examples to show that this improvement is optimal in general. These are closely related to earlier counterexamples for endpoint Strichartz estimates by Montgomery--Smith and Carbery--Hofmann. In particular, we give a negative answer to the log-removal question on the endpoint eigenfunction restriction estimates, as the log-free estimate $O(λ^{1/2})$ cannot hold in general.