Restriction estimates in a conical singular space: Schrödinger equation

Abstract

Abstract This paper continues our previous program to study the restriction estimates in a class of conical singular spaces X = C ⁢ ( Y ) = ( 0 , ∞ ) r × Y {X=C(Y)=(0,\infty)_{r}\times Y} equipped with the metric g = d ⁢ r 2 + r 2 ⁢ h {g=\mathrm{d}r^{2}+r^{2}h} , where the cross section Y is a compact ( n - 1 ) {(n-1)} -dimensional closed Riemannian manifold ( Y , h ) {(Y,h)} . Assuming the initial data possesses additional regularity in the angular variable θ ∈ Y {\theta\in 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 {\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 Schrödinger equation in this setting.