Restriction estimates in a conical singular space: wave equation

Abstract

We study the restriction estimates in a class of conical singular space $X=C(Y)=(0,\infty {)}_r\times Y$ with the metric $g=\mathrm{d}{r}^2+r^{2}h$, 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 $V=V_0{r}^{-2}$, 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.

Comments are welcome. 25 Pages