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Abstract
Let M be a connected Riemannian manifold without boundary 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))t≧0 be the heat semigroup on M. We show that the total variation of the gradient of a function u ∈ L1(M) equals the limit of the L1-norm of ∇T(t)u as t → 0. In particular, this limit is finite if and only if u is a function of bounded variation.
Received: 2006-06-15
Revised: 2006-09-29
Published Online: 2008-02-05
Published in Print: 2007-12-19
© Walter de Gruyter
