A Bourgain–Brezis–Mironescu–Dávila theorem in Carnot groups of step two

Abstract

In this note we prove the following theorem in any Carnot group of step two $\mathbb{G}$:\[\lim_{s \nearrow 1/2} (1 - 2s) \mathfrak{P}_{H,s} (E) = \frac{4}{\sqrt{\pi}} \mathfrak{P}_H (E).\]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 Dávila.