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

We study the heat equation $\frac{\partial u}{\partial t}-Δu=0,\ u(x,0)=ω(x),$ where $Δ:=dd^{*}+d^{*}d$ is the Hodge laplacian and $u(\cdot ,t)$ and $ω$ 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 $p\geq 1$ and $k\geq 0$: $\displaystyle \forall t\geq 1,\ {\left\Vert{\nabla ^{k}e^{-tΔ_{p}}}\right\Vert}_{L^{r}(M)-L^{r}(M)}\leq c(n,r,k).$ Acting on functions, i.e. with $p=0,$ we get a better result: $\displaystyle \forall k\geq 1,\ \forall t\geq 1,\ {\left\Vert{\nabla ^{k}e^{-tΔ}}\right\Vert}_{L^{r}(M)-L^{r}(M)}\leq c(n,r,k)t^{-1/2}.$

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