Kakeya-Nikodym Problems and Geodesic Restriction Estimates for Eigenfunctions
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Date
2017-05-19
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Johns Hopkins University
Abstract
We record work done by the author on the Kakeya-Nikodym problems, and we also record the joint work done by the author and Cheng Zhang on improved geodesic restriction estimates for eigenfunctions on compact Riemannian surfaces with nonpositive curvature.
The Kakeya-Nikodym problems are among the central topics in modern Harmonic analysis. The work of the author gives an alternative proof for the classical bound of Wolff for the Kakeya-Nikodym type maximal operators in Euclidean spaces R^d, d no less than 3, without appealing to the induction on scales arguments.
As a consequence of the new proof, it is also shown in that the same L^{(d+2)/2} bound holds for Nikodym maximal function for any manifold (M^d,g) with constant curvature, which generalizes Sogge's results for d=3 to any d no less than 3. As in the 3-dimensional case, we can handle manifolds of constant curvature due to the fact that, in this case, two intersecting geodesics uniquely determine a 2-dimensional totally geodesic submanifold, which allows the use of the auxiliary maximal function to reduce the problem to a 2-dimensional one.
In the joint work of the author and Cheng Zhang, we prove improved L^4 geodesic restriction estimates for eigenfunctions on compact Riemannian surfaces with nonpositive curvature. We achieve this by adapting Sogge's strategy in a recent paper. This result improves the L^4 restriction estimate of Burq, Gerard and Tzvetkov and Hu by a power of (\log\log\lambda)^{-1}. Moreover, in the special case of compact hyperbolic surfaces, we obtain further improvements in terms of (\log\lambda)^{-1} by applying the ideas from the work of Chen-Sogge and Blair-Sogge. We are able to compute various constants explicitly, by lifting calculations to the universal cover H^2.
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Keywords
Kakeya-Nikodym Problem, Eigenfunction Estimates