Revista Matemática Iberoamericana


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Volume 34, Issue 3, 2018, pp. 1071–1091
DOI: 10.4171/RMI/1016

Published online: 2018-08-27

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

Matthew D. Blair[1], G. Austin Ford[2] and Jeremy L. Marzuola[3]

(1) University of New Mexico, Albuquerque, USA
(2) AltSchool, San Francisco, USA
(3) University of North Carolina at Chapel Hill, USA

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 $C(\mathbb{S}^1_\rho) {\stackrel{\mathrm{def}}{=}} \mathbb{R}_+ \times \left(\mathbb{R} \big/ 2\pi\rho \mathbb{Z}\right)$ 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.

Keywords: Spectral clusters, polygons, conic singularities

Blair Matthew, Ford G. Austin, Marzuola Jeremy: $L^p$-bounds on spectral clusters associated to polygonal domains. Rev. Mat. Iberoamericana 34 (2018), 1071-1091. doi: 10.4171/RMI/1016