  With $\vec{\Delta}_j\geq 0$ is the uniquely determined self-adjoint
realization of the Laplace operator acting on $j$-forms on a geodesically
complete Riemannian manifold $M$ and $\nabla$ the Levi-Civita covariant
derivative, we prove amongst other things a Li-Yau type heat kernel bound for
$\nabla \mathrm{e}^{ -t\vec{\Delta}_j }$, if the curvature tensor of $M$ and
its covariant derivative are bounded, an exponentially weighted $L^p$ bound for
the heat kernel of $\nabla \mathrm{e}^{ -t\vec{\Delta}_j }$, if the curvature
tensor of $M$ and its covariant derivative are bounded, that $\nabla
\mathrm{e}^{ -t\vec{\Delta}_j }$ is bounded in $L^p$ for all $1\leq p<\infty$,
if the curvature tensor of $M$ and its covariant derivative are bounded, and a
second order Davies-Gaffney estimate (in terms of $\nabla$ and
$\vec{\Delta}_j$) for $\mathrm{e}^{ -t\vec{\Delta}_j }$ for small times, if the
$j$-th degree Bochner-Lichnerowicz potential
$V_j=\vec{\Delta}_j-\nabla^{\dagger}\nabla$ of $M$ is bounded from below (where
$V_1=\mathrm{Ric}$), which is shown to fail for large times if $V_j$ is
bounded. Based on these results, we formulate a conjecture on the boundedness
of the covariant local Riesz-transform $\nabla (\vec{\Delta}_j+\kappa)^{-1/2}$
in $L^p$ for all $1\leq p<\infty$ (which we prove for $1\leq p\leq 2$), and
explain its implications to geometric analysis, such as the
$L^p$-Calder\'on-Zygmund inequality. Our main technical tool is a Bismut
derivative formula for $\nabla \mathrm{e}^{ -t\vec{\Delta}_j }$.
