A few endpoint geodesic restriction estimates for eigenfunctions

Abstract

We prove a couple of new endpoint geodesic restriction estimates for eigenfunctions. In the case of general 3-dimensional compact manifolds, after a $TT^*$ argument, simply by using the $L^2$-boundedness of the Hilbert transform on $\R$, we are able to improve the corresponding $L^2$-restriction bounds of Burq, G\'erard and Tzvetkov and Hu. Also, in the case of 2-dimensional compact manifolds with nonpositive curvature, we obtain improved $L^4$-estimates for restrictions to geodesics, which, by H\"older's inequality and interpolation, implies improved $L^p$-bounds for all exponents $p\ge 2$. We do this by using oscillatory integral theorems of H\"ormander, Greenleaf and Seeger, and Phong and Stein, along with a simple geometric lemma (Lemma \ref{lemma3.2}) about properties of the mixed-Hessian of the Riemannian distance function restricted to pairs of geodesics in Riemannian surfaces. We are also able to get further improvements beyond our new results in three dimensions under the assumption of constant nonpositive curvature by exploiting the fact that in this case there are many totally geodesic submanifolds.