Upper bounds on derivatives of the logarithm of the heat kernel (Q1281357)

Let \(M\) be a compact, connected Riemannian manifold and denote by \(\Delta\) the Levi-Civita Laplace operator on \(M\). Let \(p_t(x,y)\) be the fundamental solution of the Cauchy initial value problem for the diffusion equation \(u_t=\frac{1}{2}\Delta u\). The main result of this paper asserts that if \(n\) is an arbitrary positive integer then the \(n\)-th derivative of the mapping \(x\longmapsto \log p_t(x,y)\) for any \(t\in(0,1)\brack\) is bounded by a constant times \(( t^{-1}+t^{-2}\text{dist}(x,y)^2)^{n/2}\).