Restriction estimates in a conical singular space: wave equation

2020/07/10 by Gao, Xiaofen, Zhang, Junyong, Zheng, Jiqiang
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2007.05161

Abstract

We study the restriction estimates in a class of conical singular space X=C(Y)=(0,∞)_r× Y with the metric g=dr²+r²h, where the cross section Y is a compact (n-1)-dimensional closed Riemannian manifold (Y,h). Let Δ_g be the Friedrich extension positive Laplacian on X, and consider the operator L_V=Δ_g+V with V=V₀r^-2, where V₀(θ)\inC^∞(Y) is a real function such that the operator Δ_h+V₀+(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+V₀+(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.

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