Some estimates related to fractal measures and Laplacians on manifolds

Abstract

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