  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.
