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.
2007
Miranda, Michele; D., Pallara; F., Paronetto; M., Preunkert
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11392/519079
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