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

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) %2> ι s * the domain of all positive powers of Δ as a function of xλ; see e.g. [9]. In works of Garding [19] and Donnelly [16], upper estimates for E(xu x2, t) (and its derivatives) were given under the assumption that M has bounded geometry. They showed that as x2 -> oo, the behavior of E{xλ, x2, t) is roughly similar to that of the e-p 2(xx,x2)/4 Euclidean heat kernel, — (p(xx, x2) denotes distance). Recall that (4ττ/) M is said to have bounded geometry if the injectivity radius i(x) of the

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