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

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(S 1 ρ ) def = R + × R 2πρZ of radius ρ > 0 equipped with the metric h(r, θ) = dr 2 + r 2 dθ 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.

Uniform resolvent estimates for critical magnetic Schrödinger operators in 2D

“…But we need to establish the Stein-Tomas restriction estimates associated with the variable coefficient operator L A,0 (perturbated by scaling critical magnetic potentials) which has its own independent interesting. Since the operator L A,0 is conically singular, we are in spirit of [3] (in which Blair-Ford-Marzuola studied the cluster estimates for a conical singular Schrödinger operator) to prove the Stein-Tomas restriction estimates. Based on the kernel of resolvent constructed in [19], due to the diffractive effect, the kernel is more singular and different from the kernel in the Euclidean setting.…”