Heat Semigroup and Functions of Bounded Variation on Riemannian Manifolds

Let M be a connected Riemannian manifold without boundary and with Ricci curvature bounded from below and such that the volume of the geodesic balls of centre x and fixed radius r>0 have a volume bounded away from 0 uniformly with respect to x, and let T(t) be the heat semigroup on M. We show that the total variation of the gradient of a summable function u equals the limit of the L^1-norm of the gradient of T(t)u as the time goes to 0. In particular, this limit is finite if and only if u is a function of bounded variation.