On the {$L^{\infty }$} norms of spectral projectors on shrinking intervals: the cases of some spheres of revolution and of the Euclidean disk

Abstract

Given a compact Riemannian surface $M$, with Laplace-Beltrami operator $\Delta$, for $\lambda >0$, let $P_{\lambda ,{\lambda }^{-\frac{1}{3}}}$ be the spectral projector on the bandwidth $[\lambda -{\lambda }^{-\frac{1}{3}},\lambda +{\lambda }^{\frac{1}{3}}]$ associated to $\sqrt{-\Delta }$. We prove a polynomial improvement on the $L^2\to L^{\infty }$ norm of $P_{\lambda ,{\lambda }^{-\frac{1}{3}}}$ for generic simple spheres of revolution (away from the poles and the equator) and for the Euclidean disk away from its center but up to the boundary. We use the Quantum Integrability of those surfaces to express the norm in terms of a joint basis of eigenfunctions for $\left(\sqrt{-\Delta },\frac{1}{i}\frac{\partial }{\partial \theta }\right)$. Then, we use that those eigenfunctions are asymptotically Lagrangian oscillatory functions, each supported on a Lagrangian torus with fold-type caustic. Thus, studying the distribution of the caustics, and using BKW decay away from the caustics, we are able to reduce the problem to counting estimates.