Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds

@article{Cheeger1982FinitePS,
  title={Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds},
  author={Jeff Cheeger and Mikhael Gromov and Michael Taylor},
  journal={Journal of Differential Geometry},
  year={1982},
  volume={17},
  pages={15-53}
}
where dEλ is the projection valued measure associated with /^Δ". A natural problem is to study the behavior of the explicit kernel kf(X)(xx, x2) representing /(/^Δ), in terms of the behavior of various geometric quantities on M. As a particularly important example we have the heat kernel E(xl9 x2, t) — ke-\2t. By use of the local parametrix and the standard elliptic estimates, one can show that for / > 0, E(xλ9 JC2, 0 is a positive (symmetric) C function of JC1? x2, t which for fixed t and (say… 

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