Let $(M,g)$ be a compact Riemannian manifold. It is known there exist Gaussian estimates of the heat kernel and its derivatives acting on functions on $M$.


The kind of estimate I'm looking for could be found in the first page of the following article : <http://www.math.uni-bielefeld.de/~grigor/grad.pdf>


My question is : it is possible to extends this kind of result at the level of the $k$-forms in $M$ ? 


More specifically I would be interested by inequalities of the following kind, for small value of $t$ :


\begin{equation\*}
||\nabla\_y H^1(x,y,t)|| \le C\_1(t) \exp({\frac{- c\_2 d(x,y)^2}{t}})
\end{equation\*}


where $H^1(x,y,t)$ is the heat kernel for $1$-forms viewed as a $(1,1)$ form on the product $M \times M \verb+\+ \Delta$, if $t$ is freeze. $\Delta$ denote the diagonal and $|| \cdot ||$ the punctual norm induced by $g$. And $C\_1(t)$ a well controlled function of $t$, some rational fraction for instance. 
I've read a lot of stuff about functions but nothing about the forms. Someone suggests me to look at the Kato's inequalities but I don't really see how to use it because this one compare the solutions of different elliptic equations, not the kernel.


Any help and references will be appreciated.



Heat Kernel estimate at the level of the form
There should be better references, but for a beginning, how about the following: Look at Lemme 1 in Thierry Bouche's paper *Convergence de la metrique de Fubini-Study d'un fibre lineaire positif*, Ann. Inst. Grenoble 1990. Here is the link to the paper:


<http://archive.numdam.org/ARCHIVE/AIF/AIF_1990__40_1/AIF_1990__40_1_117_0/AIF_1990__40_1_117_0.pdf> . 


The proof of Lemme 1 on page 122 tells you how to apply Kato's inequality to read the desired result from the leading term in the Minakshisundaram-Pleijel asymptotic expansion (the case of functions).


The paper is about complex projective manifolds, but $(M,g)$ in section 1 is an arbitrary compact Riemannian manifold. Consider just $E = M \times \mathbb{C}$ (trivial), $F = T^\*M$, $k = 1$, and $V = 0$.


