Heat kernels and function theory on metric measure spaces

It is worth observing that the Gaussian term exp ( − |x−y| 4t ) does not depend in n, whereas the other term (4πt)−n/2 reflects the dependence of the heat kernel on the underlying space via its dimension. The notion of heat kernel extends to any Riemannian manifold M. In this case, the heat kernel pt (x, y) is the minimal positive fundamental solution to the heat equation ∂u ∂t = ∆u where ∆ is the Laplace-Beltrami operator on M , and it always exists (see [11], [13], [16]). Under certain assumptions about M , the heat kernel can be estimated similarly to (1.1). For example, if M is geodesically complete and

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