Bilinear Kakeya-Nikodym averages of eigenfunctions on compact Riemannian surfaces

We obtain an improvement of the bilinear estimates of Burq, G\'erard and Tzvetkov in the spirit of the refined Kakeya-Nikodym estimates of Blair and the second author. We do this by using microlocal techniques and a bilinear version of H\"ormander's oscillatory integral theorem.Comment: 19 pages, 1 figure. Affiliation correcte

Can you hear your location on a manifold?

We introduce a variation on Kac's question, "Can one hear the shape of a drum?" Instead of trying to identify a compact manifold and its metric via its Laplace--Beltrami spectrum, we ask if it is possible to uniquely identify a point $x$ on the manifold, up to symmetry, from its pointwise counting function $$N_x(\lambda) = \sum_{\lambda_j \leq \lambda} |e_j(x)|^2,$$ where here $\Delta_g e_j = -\lambda_j^2 e_j$ and $e_j$ form an orthonormal basis for $L^2(M)$. This problem has a physical interpretation. You are placed at an arbitrary location in a familiar room with your eyes closed. Can you identify your location in the room by clapping your hands once and listening to the resulting echos and reverberations? The main result of this paper provides an affirmative answer to this question for a generic class of metrics. We also probe the problem with a variety of simple examples, highlighting along the way helpful geometric invariants that can be pulled out of the pointwise counting function $N_x$.Comment: 26 pages, 1 figur

Kakeya-Nikodym Problems and Geodesic Restriction Estimates for Eigenfunctions

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