In this note we prove the following theorem in any Carnot group of step two G: lim s↗1/2 (1 − 2s)PH,s(E) = 4 √_ PH(E). Here, PH(E) represents the horizontal perimeter of a measurable set E ⊂ G, whereas the nonlocal horizontal perimeter PH,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 Davila.
A Bourgain–Brezis–Mironescu–Dávila theorem in Carnot groups of step two
Garofalo, Nicola
Primo
;Tralli, Giulio
Ultimo
2023
Abstract
In this note we prove the following theorem in any Carnot group of step two G: lim s↗1/2 (1 − 2s)PH,s(E) = 4 √_ PH(E). Here, PH(E) represents the horizontal perimeter of a measurable set E ⊂ G, whereas the nonlocal horizontal perimeter PH,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 Davila.File in questo prodotto:
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