Restriction Estimates in a Conical Singular Space: Wave Equation

We study the restriction estimates in a class of conical singular space X= C(Y) = (0 , ∞) _r× Y with the metric g= d r²+ r²h, where the cross section Y is a compact (n- 1) -dimensional closed Riemannian manifold (Y, h). Let Δ _g be the Friedrichs extension positive Laplacian on X, and consider the operator L_V= Δ _g+ V with V= Vr^{- 2}, where V(θ) ∈ C^∞(Y) is a real function such that the operator Δ _h+ V+ (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 L_V. The smallest positive eigenvalue of the operator Δ _h+ V+ (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.