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We look at the $L^p$ bounds on eigenfunctions for polygonal domains (or moregenerally Euclidean surfaces with conic singularities) by analysis of the waveoperator 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 $h(r,\theta) = d r^2 + r^2 \, d\theta^2$. Usingexplicit oscillatory integrals and relying on the fundamental solution to thewave equation in geometric regions related to flat wave propagation anddiffraction by the cone point, we can prove spectral cluster estimatesequivalent to those in works on smooth Riemannian manifolds.

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