preprint

Inserted: 14 apr 2023

Year: 2023

Abstract:

We extend Nguyen's characterization of Sobolev spaces $W^{1,p}$ to the setting of PI-metric measure spaces such that at a.e. point the tangent space (in the Gromov-Hausdorff sense) is unique and Euclidean with a fixed dimension. We also generalize CLL14 to PI-metric measure spaces such that at a.e. point the tangent space is unique and equal to the Heisenberg group with a fixed homogeneous dimension. The approach is easier and completely different from the one in Ngu06, Ngu08.