# L p -estimates for the heat semigroup on differential forms, and related problems

## TL;DR

This paper investigates heat semigroup estimates for differential forms on certain Riemannian manifolds, establishing bounds under specific geometric conditions and exploring implications for heat kernel behavior and Riesz transforms.

## Contribution

It provides new Lp estimates for the heat semigroup on differential forms under volume doubling and Gaussian bounds, extending previous results with conditions on the Kato class.

## Key findings

- Lp bounds for heat semigroup on forms under geometric assumptions
- Gradient estimates for heat semigroup on functions
- Discussion of heat kernel bounds and Riesz transform on forms

## Abstract

We consider a complete non-compact Riemannian manifold satisfying the volume doubling property and a Gaussian upper bound for its heat kernel (on functions). Let -- $\rightarrow$ $\Delta$ k be the Hodge-de Rham Laplacian on differential k-forms with k $\ge$ 1. By the Bochner decomposition formula -- $\rightarrow$ $\Delta$ k = * + R k. Under the assumption that the negative part R -- k is in an enlarged Kato class, we prove that for all p $\in$ [1, $\infty$], e --t -- $\rightarrow$ $\Delta$ k p--p $\le$ C(t log t) D 4 (1-- 2 p) (for large t). This estimate can be improved if R -- k is strongly sub-critical. In general, (e --t -- $\rightarrow$ $\Delta$ k) t>0 is not uniformly bounded on L p for any p = 2. We also prove the gradient estimate e --t$\Delta$ p--p $\le$ Ct -- 1 p , where $\Delta$ is the Laplace-Beltrami operator (acting on functions). Finally we discuss heat kernel bounds on forms and the Riesz transform on L p for p > 2.