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

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.

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The first author is supported in part by a Progetto SID (Investimento Strategico di Dipartimento) “Non-local operators in geometry and in free boundary problems, and their connection with the applied sciences”, University of Padova, 2017. Both authors are supported in part by a Progetto SID: “Non-local Sobolev and isoperimetric inequalities”, University of Padova, 2019.

Received 17 April 2020

Accepted 14 September 2020

Published 6 December 2023