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