Consider the heat equation
$$ (\partial\_t + \Delta + V)u = 0$$
on a complete (open) Riemannian manifold with bounded geometry, where $V$ is a smooth and bounded potential.


Consider the semigroup generated by $A:=\Delta + V$ on the spaces $X = C\_0(M)$ or $X = L^p(M)$ for $1\leq p < \infty$. The domain of $A$ is in each case
$$ \mathrm{dom}(A) =\{ u \in X \mid Au \in X \},$$
where $Au$ denotes the distributional derivative. 


**My question is the following:** It is clear by parabolic regularity that $e^{-tA}u$ is smooth for any $u \in X$. But is it also true that solutions $e^{-tA}u$ satisfy 
$$ |\nabla^ku| \in X ~~~~~~~\forall ~k$$
for $u \in \mathrm{dom}(A)$?
The local statement is clear (i.e. solutions are in $W^{k, p}\_{\mathrm{loc}}(M)$ and in $C^k(M)$ for every $k$), but is it also true that we have this decay at infinity?


What about the special case of vanishing $V$?



Decay of Solutions to the Heat equation
For $C$ It follows from the interior Schauder estimates. Namely, for $r>0$ denote $H\_r(x\_0,t\_0)=\{|x-x\_0|<r^2,\ 0<t\_0-t<r\}$. Then for $0<\alpha<1$
$$
\|u\|\_{C^{2+\alpha}(H\_r(x,t))}\le C\|u\|\_{C(H\_{2r}(x,t))}.
$$
Taking $r=t/2$ for $t\le 1$ and $r=1/2$ for $t>1$ gives $|\nabla^ku| \in CM)$, $k\le2$. Differentiating the equation w.r.t. $x$ one can get analogous estimate for $|\nabla^3 u| \in C(M)$. Bootstrapping gets the required estimate for any $k$.


For the case $1<p<\infty$ the result follows from the similar theory in the anisotropic Sobolev spaces $W^{2,p}$. 


Another way is to get estimates for the fundamental solution $\Gamma$. Then denoting $u\_0=u(x,0)$ we have


$$\|\nabla^ku(\cdot,t)\|\_{L\_p}= \|\nabla^k\Gamma(\cdot,t)\*u\_0\|\_{L\_p}\le 
\|\nabla^k\Gamma(\cdot,t)\|\_{L\_1}\|u\_0\|\_{L\_p}.
$$


