# On gradient estimates for the heat kernel

## TL;DR

This paper investigates gradient estimates for the heat kernel on manifolds with controlled negative Ricci curvature and establishes boundedness of the heat operator on differential forms in various L^p spaces.

## Contribution

It provides new pointwise and L^p gradient estimates for the heat kernel on manifolds with certain negative Ricci curvature bounds and proves boundedness results for the heat operator on differential forms.

## Key findings

- Established pointwise gradient estimates under negative Ricci curvature conditions.
- Proved L^p boundedness of the heat operator on differential forms.
- Extended heat kernel analysis to manifolds with non-trivial curvature bounds.

## 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.