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Upper bounds of derivatives of the heat kernel on an arbitrary complete manifold. (English) Zbl 0842.58070
Let $$M$$ be a smooth connected complete non-compact Riemannian manifold, with corresponding Laplace operator $$\Delta$$. Let $$p(x, y, t)$$ be the heat kernel, i.e. the smallest positive fundamental solution to the heat equation $$\partial_t u - \Delta u = 0$$. If $$D > 2$$ is a real number, put $$E_n(x,t) = \int_M |\nabla^n p |^2 (x,y,t) \text{exp} \biggl({r^2 \over Dt}\biggr) dy$$. The main result of the paper is an upper bound on $$E_n$$, $$n \geq 1$$, if some upper bounds on $$E_0$$ hold. Pointwise upper bounds of $${\partial^n p\over \partial t^n}$$ are also obtained, if some pointwise upper bounds on $$p(x,y,t)$$ hold.

##### MSC:
 58J35 Heat and other parabolic equation methods for PDEs on manifolds
##### Keywords:
complete manifolds; heat kernel
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