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

Abstract

In the setting of metric measure spaces satisfying the doubling condition and the $(1,p)$-Poincar\'e 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 $N$.