$L^p$-bounds on spectral clusters associated to polygonal domains

Abstract

We look at the $L^p$ bounds on eigenfunctions for polygonal domains (or more generally Euclidean surfaces with conic singularities) by analysis of the wave operator on the flat Euclidean cone $C({\mathbb{S}}_{\rho }^1)\stackrel{\mathrm{def}}{=}{\mathbb{R}}_{+}\times \left(\mathbb{R}/2\pi \rho \mathbb{Z}\right)$ of radius $\rho >0$ equipped with the metric h$(r,\theta )=\mathrm{d}{r}^2+r^2\mathrm{d}{\theta }^2$. Using explicit oscillatory integrals and relying on the fundamental solution to the wave equation in geometric regions related to flat wave propagation and diffraction by the cone point, we can prove spectral cluster estimates equivalent to those in works on smooth Riemannian manifolds.