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 R for big balls; if there is no non-zero L harmonic 1-form, and the Ricci tensor is in L n 2 −ε ∩ L∞ for an ε > 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∆ is bounded on L for all 1 < p < ∞. Then, in presence of non-zero L harmonic 1-forms, we prove that the Riesz transform is still bounded on L for all 1 < p < n, when n > 3.