Some remarks on gradient estimates for heat kernels

Abstract

This paper is concerned with pointwise estimates for the gradient of the heat kernelKt,t>0, of the Laplace operator on a Riemannian manifoldM. Under standard assumptions onM, we show that∇Ktsatisfies Gaussian bounds if and only if it satisfies certain uniform estimates or estimates inLpfor some1≤p≤∞. The proof is based on finite speed propagation for the wave equation, and extends to a more general setting. We also prove that Gaussian bounds on∇Ktare stable under surjective, submersive mappings between manifolds which preserve the Laplacians. As applications, we obtain gradient estimates on covering manifolds and on homogeneous spaces of Lie groups of polynomial growth and boundedness of Riesz transform operators.