Sharp endpoint estimates for eigenfunctions restricted to submanifolds of codimension 2

TL;DR

This paper proves that the log loss in endpoint eigenfunction restriction estimates for submanifolds of codimension 2 can be removed in higher dimensions for totally geodesic submanifolds and in 3D for certain curves, advancing understanding of eigenfunction behavior.

Contribution

It extends the removal of the log loss at endpoint estimates to higher dimensions and more general submanifolds, building on prior results for geodesics in 3D.

Findings

Log loss removed for totally geodesic submanifolds of codimension 2 in higher dimensions.

Log loss removed for curves with nonvanishing geodesic curvature in 3D.

Connections established with singular oscillatory integrals and Hilbert transforms along curves.

Abstract

Burq-G\'erard-Tzvetkov and Hu established $L^p$ estimates ($2\le p\le \infty$) for the restriction of eigenfunctions to submanifolds. The estimates are sharp, except for the log loss at the endpoint $L^2$ estimates for submanifolds of codimension 2. It has long been believed that the log loss at the endpoint can be removed in general, while the problem is still open. So this paper is devoted to the study of sharp endpoint restriction estimates for eigenfunctions in this case. Chen and Sogge removed the log loss for the geodesics on 3-dimensional manifolds. In this paper, we generalize their result to higher dimensions and prove that the log loss can be removed for totally geodesic submanifolds of codimension 2. Moreover, on 3-dimensional manifolds, we can remove the log loss for curves with nonvanishing geodesic curvatures, and more general finite type curves. The problem in 3D is essentially related to Hilbert transforms along curves in the plane and a class of singular oscillatory integrals studied by Phong-Stein, Ricci-Stein, Pan, Seeger, Carbery-P\'erez.