论文标题
圆锥形奇异空间中的限制估计：波方程
Restriction estimates in a conical singular space: wave equation
论文作者
论文摘要
We study the restriction estimates in a class of conical singular space $X=C(Y)=(0,\infty)_r\times Y$ with the metric $g=\mathrm{d}r^2+r^2h$, where the cross section $Y$ is a compact $(n-1)$-dimensional closed Riemannian manifold $(Y,h)$.令$Δ_g$为$ x $上的Friedrich Extension阳性laplacian，并考虑操作员$ \ Mathcal {l} _V =δ_g+v $ with $ v = v_0r^{ - 2} $，其中$ v_0（θ） $Δ_H+V_0+（N-2）^2/4 $是正的。在本文中，我们证明了与$ \ Mathcal {l} _v $相关的波方程解决方案的修改限制估计。操作员的最小正征值$Δ_H+V_0+（N-2）^2/4 $在结果中起着重要作用。 作为一种应用，对于与利益无关的应用，我们证明了在这种情况下的波动方程的局部能量估计和龙骨 - 史密斯估计。
We study the restriction estimates in a class of conical singular space $X=C(Y)=(0,\infty)_r\times Y$ with the metric $g=\mathrm{d}r^2+r^2h$, 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 $\mathcal{L}_V=Δ_g+V$ with $V=V_0r^{-2}$, where $V_0(θ)\in\mathcal{C}^\infty(Y)$ is a real function such that the operator $Δ_h+V_0+(n-2)^2/4$ is positive. In the present paper, we prove a type of modified restriction estimates for the solutions of wave equation associated with $\mathcal{L}_V$. The smallest positive eigenvalue of the operator $Δ_h+V_0+(n-2)^2/4$ 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.
