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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.

user96849
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  • Is there a definition of $K$ in a local coordinate $\phi$ ? – reuns Jul 28 '17 at 14:51
  • @reuns Unfortunately it is not the case. – user96849 Jul 28 '17 at 19:07
  • Did you try the "dirty" way and prove the statement using just the definition of the heat kernel? (E.g. the definition from Chavel's book "Eigenvalues in Riemannian Geometry" Chapter VI.4, p. 153, equation (42).) This would be my first approach. Other than that you could try to use the fact that the heat kernel is "almost euclidean" (see e.g. Chavel, p. 159). – user12390 Jul 30 '17 at 19:18
  • @user12390 No, I did not. But I could try this "dirty" way. – user96849 Jul 31 '17 at 11:46

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