Let $(M,g)$ an $N$-dimensional $(N\geq 1)$ compact (no boundary) Riemannian manifold. Let $K(t,x,y)$ its heat kernel.
Do the following estimates hold true?
$\exists C=C(M,g)>0$ constant such that for all $t>0$ and all $x,y\in M$ $$ |\partial_t^n\nabla^m_xK(t,x,y)|_g\leq Ct^{-(N+2n+m)/2} $$ for $2n+m\leq 3$, where $\partial_t^n$ is the $n$-th time derivative and $\nabla^m_x$ is the $m$-th covariant $x$ derivative.
Where could I find a reference?
Many thanks.