Bourgain-Brezis-Mironescu approach in metric spaces with Euclidean tangents

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

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.

与分数周长相关的几何不等式：度量测度空间中的分数Poincaré、等周与boxing不等式

Geometric inequalities related to fractional perimeter: fractional Poincaré, isoperimetric, and boxing inequalities in metric measure spaces

本文在支撑$(1,1)$-Poincaré不等式的度量测度空间中建立了带缩放常数$(1-\theta)$的分数Poincaré不等式，并由此导出分数等周与boxing不等式，进而刻画了Poincaré不等式和下Ahlfors正则性。