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A Few Endpoint Geodesic Restriction Estimates for Eigenfunctions

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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 \({\mathbb{R}}\) , we are able to improve the corresponding L 2-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 L 4-estimates for restrictions to geodesics, which, by Hölder’s inequality and interpolation, implies improved L p-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.

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Authors and Affiliations

  1. Department of Mathematics, Johns Hopkins University, Baltimore, MD, 21218, USA

    Xuehua Chen & Christopher D. Sogge

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  1. Xuehua Chen
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  2. Christopher D. Sogge
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Correspondence to Christopher D. Sogge.

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Communicated by S. Zelditch

The authors were supported in part by the NSF grant DMS-1069175.

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Chen, X., Sogge, C.D. A Few Endpoint Geodesic Restriction Estimates for Eigenfunctions. Commun. Math. Phys. 329, 435–459 (2014). https://doi.org/10.1007/s00220-014-1959-3

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