Two characterization of \(BV\) functions on Carnot groups via the heat semigroup

Michele Miranda

(University of Ferrara)

Event: ERC Workshop on Geometric Analysis on sub-Riemannian and Metric Spaces

Date: Oct 13, 2011, time: 17:50

Place: Centro De Giorgi, Scuola Normale Superiore

Abstract. In this talk I shall present a recent paper, joint work with M.Bramanti and D.Pallara, accepted for publication on International Mathematics Research Notices. The aim of the paper is to provide two different characterizations of sets with finite perimeter and functions of bounded variation in Carnot groups, analogous to those which hold in Euclidean spaces, in terms of the short-time behavior of the heat semigroup. The first characterization follows the original definition given by De Giorgi of functions with bounded variation; if we denote by \((T_t)_{t\geq 0}\) the heat semigroup associated to the sub--Laplacian, then we investigate the behavior as \(t\to 0\) of the map
\[ f(t)=\int \vert \nabla T_t u \vert . \]
In the original paper of De Giorgi it was proved that \(f\) is a monotone map and that
\[ \lim_{t\to 0} f(t)<+\infty \]
if and only if \(u\) has bounded variation; in this case the limit coincides with the total variation of \(u\). In the Carnot group setting, we are not able to prove the existence of the limit, but we show that \(f(t)\) is bounded if and only if \(u\) has finite total variation, with total variation quantitatively controlled by the small time behavior of \(f(t)\).

The second characterization was suggested to us by a paper of Ledoux and holds under the hypothesis that the reduced boundary of a set of finite perimeter is rectifiable, a result that presently is known in Step 2 Carnot groups. In this case we investigate, when reducing the study on Borel sets \(E\), the function

\[ g(t)=\frac{1}{\sqrt{t}} \int_{E^c} T_t\chi_E. \]

It is shown that

\[ \lim_{t\to 0} g(t) \]

exist and it is finite if and only if \(E\) has finite perimeter and this limit is given by a weighted,
possibly anisotropic, perimeter.