A Bourgain-Brezis-Mironescu-D\'avila theorem in Carnot groups of step two

In this note we prove the following theorem in any Carnot group of step two $\mathbb{ G}$: \[ \underset{ s\nearrow 1/2}{ \lim} (1 - 2s) \mathfrak P_{ H,s}(E) = \frac{ 4}{ \sqrt \pi}\ \mathfrak P_H(E). \] Here, $\mathfrak P_H(E)$ represents the horizontal perimeter of a measurable set $E\subset \mathbb{ G}$, whereas the nonlocal horizontal perimeter $\mathfrak P_{ H,s}(E)$ is a heat based Besov seminorm. This result represents a dimensionless sub-Riemannian counterpart of a famous characterisation of Bourgain-Brezis-Mironescu and D\'avila.

Categories: math.AP