Strichartz estimates and wave equation in a conic singular space

Abstract

Consider the metric cone $X=C(Y)=(0,\infty)_r\times Y$ with the metric $g=\mathrm{d}r^2+r^2h$ where the cross section $Y$ is a compact $(n-1)$-dimensional Riemannian manifold $(Y,h)$. Let $\Delta_g$ be the Friedrich extension positive Laplacian on $X$ and let $\Delta_h$ be the positive Laplacian on $Y$, and consider the operator $\LL_V=\Delta_g+V_0 r^{-2}$ where $V_0\in\CC^\infty(Y)$ such that $\Delta_h+V_0+(n-2)^2/4$ is a strictly positive operator on $L^2(Y)$. In this paper, we prove the global-in-time Strichartz estimates without loss for the wave equation associated with the operator $\LL_V$ which verifies\cite[Remark 2.4]{wang} Wang's conjecture for wave equation. The range of the admissible pair is sharp and is influenced by the smallest eigenvalue of $\Delta_h+V_0+(n-2)^2/4$. To prove the result, we show a Sobolev inequality and a boundedness of a generalized Riesz transform in this setting. In addition, as an application, we study the well-posed theory and scattering theory for energy-critical wave equation with small data on this setting of dimension $n\geq3$.