Restriction estimates in a conical singular space: Schrodinger equation

This paper continues our previous program to study the restriction estimates in a class of conical singular spaces X = C( Y) = ( 0, infinity)(r) x Y equipped with the metric g = dr(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 theta is an element of Y, we prove some linear restriction estimates for the solutions of Schrodinger equations on the cone X. The smallest positive eigenvalue of the operator Delta(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 Schrodinger equation in this setting.