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A few endpoint geodesic restriction estimates for eigenfunctions. (English) Zbl 1293.58011

Summary: We prove a couple of new endpoint geodesic restriction estimates for eigenfunctions. In the case of general 3-dimensional compact manifolds, after a \(TT^\ast\) 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 N. Burq et al. [Duke Math. J. 138, No. 3, 445–486 (2007; Zbl 1131.35053)] and R. Hu [Forum Math. 21, No. 6, 1021–1052 (2009; Zbl 1187.35147)]. 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 \(p\geq 2\). We do this by using oscillatory integral theorems of L. Hörmander [Ark. Mat. 11, 1–11 (1973; Zbl 0254.42010)], A. Greenleaf and A. Seeger [J. Reine Angew. Math. 455, 35–56 (1994; Zbl 0799.42008)] and D. H. Phong and E. M. Stein [Int. Math. Res. Not. 1991, No. 4, 49–60 (1991; Zbl 0761.46033)], 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.

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
35P20 Asymptotic distributions of eigenvalues in context of PDEs
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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