A Gaussian estimate for the heat kernel on differential forms and application to the Riesz transform

TL;DR

This paper establishes Gaussian bounds for the heat kernel on 1-forms on certain Riemannian manifolds and demonstrates the boundedness of the Riesz transform on L^p spaces under these geometric conditions.

Contribution

It provides new Gaussian estimates for the heat kernel on 1-forms and applies these results to prove the boundedness of the Riesz transform on L^p spaces for manifolds with specific geometric properties.

Findings

Gaussian estimate on the heat kernel for 1-forms

Boundedness of the Riesz transform on L^p spaces

Conditions on Ricci tensor and volume growth for these results

Abstract

Let $(M^m,g)$ be a m-dimensional complete Riemannian manifold which satisfies the n-Sobolev inequality and on which the volume growth is comparable to the one of $\R^n$ for big balls; if the Hodge Laplacian on 1-forms is strongly positive and the Ricci tensor is in $L^{\frac{n}{2}\pm \epsilon}$ for an $\epsilon>0$, then we prove a Gaussian estimate on the heat kernel of the Hodge Laplacian on 1-forms. This allows us to prove that, under the same hypotheses, the Riesz transform $d\Delta^{-1/2}$ is bounded on $L^p$ for all $1