I am interested in Gaussian-type upper bounds for the heat kernel on forms in a compact Riemannian manifold $M$ of dimension $n$. My understanding is that the Sobolev embedding $H^1\hookrightarrow L^{2n/(n-2)}$ holds for compact manifolds, and so there is a bound of the form $$ p(t,x,y) \leq C t^{-n/2} \exp( - \frac{d^2(x,y)}{Ct}), $$ where $p(t,x,y)$ is the heat kernel.
I am curious about the situation for differential forms (so, the heat kernel for the Hodge laplacian). In this case, the heat kernel $p(t,x,y)$ is a tensor, and I imagine that there is an inequality $$ |p(t,x,y)| \leq C t^{-n/2} \exp( - \frac{d^2(x,y)}{Ct}). $$
However, the only references I can easily find treat the seemingly more difficult case of a non-compact Riemannian manifold. Is there a good reference for the above inequality (if it is true!)?