Gradient estimates for the heat semigroup on forms in a complete Riemannian manifold

Abstract

We study the heat equation $\frac{\partial u}{\partial t}-\Delta u=0,u(x,0)=\omega (x),$ where $\Delta :=d{d}^{\ast }+d^{\ast }d$ is the Hodge laplacian and $u(\cdot ,t)$ and $\omega$ are $p$-differential forms in the complete Riemannian manifold $(M,g).$ Under weak bounded geometrical assumptions we get estimates on its semigroup of the form: acting on $p$-forms with $p\ge 1$ and $k\ge 0$: $\forall t\ge 1,{\Vert {\nabla }^k{e}^{-t{\Delta }_p}\Vert }_{L^r(M)-L^r(M)}\le c(n,r,k).$ Acting on functions, i.e. with $p=0,$ we get a better result: $\forall k\ge 1,\forall t\ge 1,{\Vert {\nabla }^k{e}^{-t\Delta }\Vert }_{L^r(M)-L^r(M)}\le c(n,r,k)t^{-1/2}.$

we correct some mistakes and modify the presentation