Heat Kernel Estimate in a Conical Singular Space

Let (X, g) be a product cone with the metric g= dr²+ r²h , where X= C(Y) = (0 , ∞) _r× 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= - Δ _g+ Vr^{- 2} , where - Δ _g is the positive Friedrichs extension Laplacian on X and V= V(y) r^{- 2} and V∈ C^∞(Y) is a real function such that the operator - Δ _h+ V+ (n- 2) ²/ 4 is a strictly positive operator on L²(Y) . The new ingredient of the proof is the Hadamard parametrix and finite propagation speed of wave operator on Y.