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

Let $(M,g)$ be a complete Riemannian manifold which satisfies a Sobolev inequality of dimension $n$ , and on which the volume growth is comparable to the one of ${\mathbb{R }}^n$ for big balls; if there is no non-zero $L^2$ harmonic 1-form, and the Ricci tensor is in $L^{\frac{n}{2}-\varepsilon }\cap L^\infty $ for an $\varepsilon >0$ , then we prove a Gaussian estimate on the heat kernel of the Hodge Laplacian acting on 1-forms. This allows us to prove that, under the same hypotheses, the Riesz transform $d\varDelta ^{-1/2}$ is bounded on $L^p$ for all $1<p<\infty $ . Then, in presence of non-zero $L^2$ harmonic 1-forms, we prove that the Riesz transform is still bounded on $L^p$ for all $1<p<n$ , when $n>3$ .