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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.

58J35 Heat and other parabolic equation methods for PDEs on manifolds
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