| Résume | I will talk about the problems of L^q norm bounds for Laplace eigenfunctions with large eigenvalues on compact Riemannian manifolds. The classical local bound was proved by Sogge, which is sharp on spheres but is expected to be improved with some negative curvature assumption. For Hecke-Maass forms on compact arithmetic hyperbolic surfaces, Iwaniec and Sarnak used the method of arithmetic amplification to obtain a power saving for the sup-norm bound. We may bound the L^6 norms of Hecke-Maass forms using their sup-norms and microlocal L^6 Kakeya-Nikodym norms, which measure how they concentrate around points and around geodesics, respectively. Using this method, we obtain some power saving for the L^6 norm bound by combining the sup-norm bound of Iwaniec-Sarnak and a new improved microlocal Kakeya-Nikodym estimate. This is based on recent joint work with Xiaoqi Huang. |
