Abstract and Applied Analysis
Volume 2006 (2006), Article ID 73020, 10 pages

Some remarks on gradient estimates for heat kernels

Nick Dungey

School of Mathematics, the University of New South Wales, Sydney 2052, Australia

Received 27 September 2004; Accepted 1 March 2005

Copyright © 2006 Nick Dungey. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 1p. 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.