Let Δ be the Laplace-Beltrami operator on an$n$dimensional complete$C$^{∞}manifold$M$In this paper we establish an estimate of$e$^{$tΔ$}$(dμ)$valid for all$t$>0 where$dμ$is a locally uniformly α dimensional measure on$M$0≤α≤$n$The result is used to study the mapping properties of ($I$-$t$Δ)^{-β}considered as an operator from$L$^{$p$}$(M dμ)$to$L$^{$p$}$(M dx)$where$dx$is the Riemannian measure on$M β>(n−α)/2p′ 1/p+1/p′=1 1≤p≤∞$