On gradient estimates for the heat kernel

Abstract

We study pointwise and $L^p$ gradient estimates of the heat kernel, on manifolds that may have some amount of negative Ricci curvature, provided it is not too negative (in an integral sense) at infinity. We also prove uniform boundedness results on $L^p$ spaces for the heat operator of the Hodge Laplacian on differential forms.

67 pages, a mistake in the proof of Corollary 1.9 has been corrected; improved presentation of the proofs of Theorems 1.11 and 1.14