# Heat Kernel estimate at the level of the form

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.

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 $V = 0$.