Oscillatory Integrals and Eigenfunction Restriction Estimates

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Johns Hopkins University
The thesis consists of two parts. In the first part, we prove optimal L^2 estimates of the restrictions to a family of curves of the eigenfunctions on general compact smooth 3-dimensional Riemannian manifolds. This family includes geodesics, smooth curves with nonvanishing geodesic curvatures, and those curves satisfying certain finite-type condition, such as all the analytic curves on analytic manifolds. These results sharpen the corresponding bounds of Burq, G\'erard and Tzvetkov, Hu, Chen and Sogge. We show that the problem is essentially related to Hilbert transforms along curves in the plane and a class of singular oscillatory integrals studied by Phong and Stein, Ricci and Stein, Pan, Seeger, Carbery and P\'erez, Nagel and Wainger. In the second part of the thesis, we show that one can obtain logarithmic improvements of L^2 geodesic restriction estimates for eigenfunctions on 3-dimensional compact Riemannian manifolds with constant negative curvature. We obtain a (\log\lambda)^{-\frac12} gain for the L^2-restriction bounds, which improves the corresponding bounds of Burq, G\'erard and Tzvetkov, Hu, Chen and Sogge. We achieve this by adapting the approaches developed by Chen and Sogge, Blair and Sogge, Xi and the author. We derive an explicit formula for the wave kernel on 3D hyperbolic space, which improves the kernel estimates from the Hadamard parametrix in Chen and Sogge. We prove detailed oscillatory integral estimates with fold singularities by Phong and Stein and use the Poincar\'e half-space model to establish bounds for various derivatives of the distance function restricted to geodesic segments on the universal cover H^3.
Oscillatory integral, eigenfunction