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 \(g=\mathrm {d}r^2+r^2h\), where the cross section Y is a compact \((n-1)\)-dimensional closed Riemannian manifold (Y, h). Let \(\Delta _g\) be the Friedrichs 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.