On the asymptotic behaviour of the fractional Sobolev seminorms in metric measure spaces: Bourgain-Brezis-Mironescu's theorem revisited

2021/10/12 by Bang-Xian Han, Han, Bang-Xian, Andrea Pinamonti +1
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG) #math.FA #math.MG

paper · pdf · doi:10.48550/arxiv.2110.05980

arxiv created 2021/10/12 · arxiv updated 2021/10/13

Abstract

We generalize Bourgain-Brezis-Mironescu's asymptotic formula for fractional Sobolev functions, in the setting of abstract metric measure spaces, under the assumption that at almost every point the tangent space in the measured Gromov-Hausdorff sense is a finite dimensional Banach space or a Carnot group. Our result not only covers the known results concerning Euclidean spaces, weighted Riemannian manifolds and finite dimensional Banach spaces, but also extends Bourgain-Brezis-Mironescu's formula to \rcdkn spaces and sub-Riemannian manifolds.

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