On gradient estimates for heat kernels

Abstract

We study pointwise and Lp gradient estimates of the heat kernels of both the scalar Laplacian, as well as the Hodge Laplacian on k-forms, on manifolds that may have some amount of negative Ricci curvature, provided it is not too negative (in an integral sense) at infinity. Such heat kernel estimates have already been obtained by the author, together with Coulhon and Sikora, provided certain L 2-cohomology spaces are trivial. This is however a strong topological assumption, and it is desirable to weaken it. The main point of the current work is to investigate what happens when these L 2-cohomology spaces are non-trivial. We find that the answer depends on some Lq integrability properties of L 2-harmonic forms.