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