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

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 ${\mathbb{R}}^n$ for big balls; if the Hodge Laplacian on 1-forms is strongly positive and the Ricci tensor is in $L^{\frac{n}{2}\pm ϵ}$ for an $ϵ>0$, 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 $d{\mathrm{\Delta }}^{-1\mathrm{/}2}$ is bounded on $L^p$ for all $1