Let M be a connected Riemannian manifold 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 function u in L^1 equals the limit of the L^1 norm of T(t)u as t goes to 0. In particular, this limit is finite if and only if u is a function of bounded variation.
Heat Semigroup and Functions of Bounded Variation on Riemannian Manifolds
MIRANDA, Michele;
2007
Abstract
Let M be a connected Riemannian manifold 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 function u in L^1 equals the limit of the L^1 norm of T(t)u as t goes to 0. In particular, this limit is finite if and only if u is a function of bounded variation.File in questo prodotto:
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