Restriction estimates in a conical singular space: wave equation

TL;DR

This paper establishes modified restriction estimates for wave equations on conical singular spaces with potential, revealing how the geometry and spectral properties influence wave behavior and energy estimates.

Contribution

It introduces new restriction estimates for wave equations on conical singular spaces with potential, linking spectral data to wave behavior.

Findings

Restriction estimates depend on the smallest eigenvalue of a related operator.

Established local energy decay estimates in conical singular spaces.

Derived Keel-Smith-Sogge estimates for wave equations in this setting.

Abstract

We study the restriction estimates in a class of conical singular space $X=C(Y)=(0,\infty)_r\times Y$ with the metric $g=\mathrm{d}r^2+r^2h$, where the cross section $Y$ is a compact $(n-1)$-dimensional closed Riemannian manifold $(Y,h)$. Let $\Delta_g$ be the Friedrich extension positive Laplacian on $X$, and consider the operator $\mathcal{L}_V=\Delta_g+V$ with $V=V_0r^{-2}$, where $V_0(\theta)\in\mathcal{C}^\infty(Y)$ is a real function such that the operator $\Delta_h+V_0+(n-2)^2/4$ is positive. In the present paper, we prove a type of modified restriction estimates for the solutions of wave equation associated with $\mathcal{L}_V$. The smallest positive eigenvalue of the operator $\Delta_h+V_0+(n-2)^2/4$ plays an important role in the result. As an application, for independent of interests, we prove local energy estimates and Keel-Smith-Sogge estimates for the wave equation in this setting.