The Estimate of Higher Derivatives of Logarithmic Heat Kernel on Compact Riemannian Manifold

Let p_M(t,x,y) be the minimal heat kernel of a d-dimenional compact Riemannian manifold M for any time t\in(0,1 and x,y\in M. Using the horizontal Brown bridge on M, we prove that, for any nonnegative integers n and m, there is a constant C depending on n,m and the manifold M, such that |\nabla^n_x\nabla^m_y\ln p_M(t,x,y)|\leq Cd(x,y)/t+1/\sqrtt\,^n+m, which generalizes the conclusion of the higher derivatives of the logarithmic heat kernel \ln p_M(t,x,y) about single variable in \ncite1.