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Let $(M^m,g)$ be a m-dimensional complete Riemannian manifold which satisfiesthe n-Sobolev inequality and on which the volume growth is comparable to theone of $\R^n$ for big balls; if the Hodge Laplacian on 1-forms is stronglypositive 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 HodgeLaplacian 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

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