Some remarks on gradient estimates for heat kernels

This paper is concerned with pointwise estimates for the gradientof the heat kernel Kt, t>0, of the Laplace operator on aRiemannian manifold M. Under standard assumptions on M, we show that ∇Kt satisfies Gaussian bounds if and only if itsatisfies certain uniform estimates or estimates in Lp for some 1≤p≤∞. The proof is based on finite speedpropagation for the wave equation, and extends to a more generalsetting. We also prove that Gaussian bounds on ∇Kt are stable under surjective, submersive mappings between manifoldswhich preserve the Laplacians. As applications, we obtain gradientestimates on covering manifolds and on homogeneous spaces of Liegroups of polynomial growth and boundedness of Riesz transformoperators