Restriction estimates in a conical singular space: Schrödinger equation.

This paper continues our previous program to study the restriction estimates in a class of conical singular spaces X = C ⁢ (Y) = (0 , ∞) r × Y equipped with the metric g = d ⁢ r 2 + r 2 ⁢ h , where the cross section Y is a compact (n - 1) -dimensional closed Riemannian manifold (Y , h) . Assuming the initial data possesses additional regularity in the angular variable θ ∈ Y , we prove some linear restriction estimates for the solutions of Schrödinger equations on the cone X. The smallest positive eigenvalue of the operator Δ h + V 0 + (n - 2) 2 / 4 plays an important role in the result. As applications, we prove local energy estimates and Keel–Smith–Sogge estimates for the Schrödinger equation in this setting. [ABSTRACT FROM AUTHOR]