  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. 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 holomorphic function in
a suitable strip of the complex plane, and satisfies Mihlin-Hormander type
conditions of appropriate order at infinity, then the operator m(L) 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.
