Some remarks on gradient estimates for heat kernels

This paper is concerned with pointwise estimates for the gradient of the heat kernel K t , t > 0 , of the Laplace operator on a Riemannian manifold M . Under standard assumptions on M , we show that ∇ K t satisfies Gaussian bounds if and only if it satisfies certain uniform estimates or estimates in L p for some 1 ≤ 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 ∇ K t are 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.