• DOI:10.4171/rmi/1016
• Corpus ID: 15191006

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

@article{Blair2015LpboundsOS,
  title={\$L^p\$-bounds on spectral clusters associated to polygonal domains},
  author={Matthew D. Blair and G. Austin Ford and Jeremy Louis Marzuola},
  journal={Revista Matem{\'a}tica Iberoamericana},
  year={2015}
}

• Matthew D. Blair, G. A. Ford, J. Marzuola
• Published 1 April 2015
• Mathematics
• Revista Matemática Iberoamericana

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.