Bourgain\u2013Brezis\u2013Mironescu Approach in Metric Spaces with Euclidean Tangents

Abstract

In the setting of metric measure spaces satisfying the doubling condition and the (1, p)-Poincaré inequality, we prove a metric analogue of the Bourgain–Brezis–Mironescu formula for functions in the Sobolev space W^{1,p}(X,d,nu ), under the assumption that for nu -a.e. point the tangent space in the Gromov–Hausdorff sense is Euclidean with fixed dimension N.