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: $\displaystyle \forall k\geq 1,\ \forall t\geq 1,\ { \left\Vert{ \nabla ^{ k}e^{ -t\Delta }}\right\Vert}_{ L^{ r}(M)-L^{ r}(M)}\leq c(n,r,k)t^{ -1/2}.$
Categories: math.AP