Estimates for functions of the laplacian on manifolds with bounded geometry

In this paper we consider a complete connected noncompact Riemannian manifold M with Ricci curvature bounded from below and positive injectivity radius. Denote by L the Laplace–Beltrami operator on M and by D the operator (L − b)^{1/2}, where b denotes the bottom of the spectrum of L. We assume that the kernel associated to the heat semigroup generated by L satisfies a mild decay condition at infinity. We prove that if m is a bounded, even holomorphic function in a suitable strip of the complex plane, and satisfies Mihlin–H¨ormander type conditions of appropriate order at infinity, then the operator m(D) extends to an operator of weak type 1. This partially extends a celebrated result of J. Cheeger, M. Gromov and M. Taylor, who proved similar results under much stronger curvature assumptions on M, but without any assumption on the decay of the heat kernel.