A Few Endpoint Geodesic Restriction Estimates for Eigenfunctions

By Xuehua Chen, Christopher D. Sogge. 2012

47 citations recorded by the source corpus.

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 L2-boundedness of the Hilbert transform on $${\mathbb{R}}$$R , we are able to improve the corresponding L2-restriction bounds of Burq, Gérard and Tzvetkov (Duke Math J 138:445–486, 2007) and Hu (Forum Math 6:1021–1052, 2009). Also, in the case of 2-dimensional compact manifolds with nonpositive curvature, we obtain improved L4-estimates for restrictions to geodesics, which, by Hölder’s inequality and interpolation, implies improved Lp-bounds for all exponents p ≥ 2. We do this by using oscillatory integral theorems of Hörmander (Ark Mat 11:1–11, 1973), Greenleaf and Seeger (J Reine Angew Math 455:35–56, 1994) and Phong and Stein (Int Math Res Notices 4:49–60, 1991), along with a simple geometric lemma (Lemma 3.2) 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.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/4783c8b4aee56c75bf69dd9eb7bd48dab31c495d

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In Applied Mathematics

Citations in the corpus

Concerning Toponogov’s theorem and logarithmic improvement of estimates of eigenfunctions (2015, background): “Chen and one of us [10] also showed that when n = 3 one could improve on the universal bounds (saturated in this case by both zonal harmonics and highest weight spherical harmonics) assuming that (M, g) is of constant nonpositive curvature.”
Improvement of eigenfunction estimates on manifolds of nonpositive curvature (2012, background): “In related work Sogge and Zelditch [11] and Chen and Sogge [6] study improvements to geodesic restriction theorems for low p in the case of nonpositive curvature.”