紧致黎曼流形上对数热核的高阶导数估计

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

摘要: 设M为一个d,维紧致黎曼流形,对任意的t\in(0,1,x,y\in M, 记p_M(t,x,y)是M的极小热核.本文利用流形M上的水平布朗桥, 把文献\ncite1中关于对数热核\ln p_M(t,x,y)的单变量的高阶导数估计推广到关于(x,y)两个变量上,即对于任意的非负整数n,m, 都存在依赖于n,m和流形M的常数C使得下式成立:|\nabla^n_x\nabla^m_y\ln p_M(t,x,y)|\leq Cd(x,y)/t+1/\sqrtt\,^n+m.

Abstract: 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.