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

Baptiste Devyver (1)

1 LMJL - Laboratoire de Mathématiques Jean Leray

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

Mots clés: Heat Kernel; Gaussian estimate; Schrodinger operator; Hodge Laplacian; gradient estimates

Domaines: Géométrie différentielle [math.DG]; Equations aux dérivées partielles [math.AP]; Théorie spectrale [math.SP]; Analyse fonctionnelle [math.FA]

Baptiste Devyver. A Gaussian estimate for the heat kernel on differential forms and application to the Riesz transform. Mathematische Annalen, 2014, 358 (1-2), pp.25-68. ⟨hal-00538547v3⟩