Heat kernel estimate in a conical singular space

Abstract

Let $(X,g)$ be a product cone with the metric $g=d{r}^2+r^{2}h$, where $X=C(Y)=(0,\infty {)}_r\times Y$ and the cross section $Y$ is a $(n-1)$-dimensional closed Riemannian manifold $(Y,h)$. We study the upper boundedness of heat kernel associated with the operator $L_V=-{\Delta }_g+V_0{r}^{-2}$, where $-{\Delta }_g$ is the positive Friedrichs extension Laplacian on $X$ and $V=V_0(y)r^{-2}$ and $V_0\in {\mathcal{C}}^{\infty }(Y)$ is a real function such that the operator $-{\Delta }_h+V_0+(n-2{)}^2/4$ is a strictly positive operator on $L^2(Y)$.The new ingredient of the proof is the Hadamard parametrix and finite propagation speed of wave operator on $Y$.