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

TL;DR

This paper establishes $L^p$ bounds on eigenfunctions for polygonal domains and Euclidean surfaces with conic singularities by analyzing wave operators and explicit oscillatory integrals, extending spectral cluster estimates to non-smooth geometries.

Contribution

It provides new $L^p$ spectral cluster bounds for polygonal domains using wave analysis and oscillatory integrals, generalizing results from smooth to singular geometries.

Findings

Spectral cluster estimates are extended to polygonal domains.

Explicit oscillatory integral techniques are used for analysis.

Results match those known for smooth Riemannian manifolds.

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 $C(\mathbb{S}^1_\rho) := \mathbb{R}_+ \times \left(\mathbb{R} \big/ 2\pi\rho \mathbb{Z}\right)$ of radius $\rho > 0$ equipped with the metric $h(r,\theta) = d r^2 + r^2 \, d\theta^2$. 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.