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Gradient estimates for the heat semigroup on forms in a complete Riemannian manifold (2003.03985v2)
Published 9 Mar 2020 in math.AP
Abstract: 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: $\displaystyle \forall k\geq 1,\ \forall t\geq 1,\ {\left\Vert{\nabla {k}e{-t\Delta }}\right\Vert}_{L{r}(M)-L{r}(M)}\leq c(n,r,k)t{-1/2}.$