Abstract

We study pointwise and L^p 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 L²-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 L²-cohomology spaces are non-trivial. We find that the answer depends on some L^q integrability properties of L²-harmonic forms.

摘要

我们研究了标量拉普拉斯算子和霍奇拉普拉斯算子在k 型上的热核的逐点和L ^p梯度估计，在可能具有一定量负 Ricci 曲率的流形上，只要它不太负（在一个整数意义）在无穷大。如果某些L ^{2 上同}调空间是微不足道的，那么作者已经与 Coulhon 和 Sikora 一起获得了这样的热核估计。然而，这是一个强拓扑假设，需要削弱它。当前工作的重点是研究当这些L ^{2 -上同}调空间非平凡时会发生什么。我们发现答案取决于一些L ^qL ^{2 -}谐波形式的可积性。