English

Testing the Sobolev property with a single test plan

Functional Analysis 2020-06-09 v1 Metric Geometry

Abstract

We prove that in a vast class of metric measure spaces (namely, those whose associated Sobolev space is separable) the following property holds: a single test plan can be used to recover the minimal weak upper gradient of any Sobolev function. This means that, in order to identify which are the exceptional curves in the weak upper gradient inequality, it suffices to consider the negligible sets of a suitable Borel measure on curves, rather than the ones of the pp-modulus. Moreover, on RCD\sf RCD spaces we can improve our result, showing that the test plan can be also chosen to be concentrated on an equi-Lipschitz family of curves.

Keywords

Cite

@article{arxiv.2006.03628,
  title  = {Testing the Sobolev property with a single test plan},
  author = {Enrico Pasqualetto},
  journal= {arXiv preprint arXiv:2006.03628},
  year   = {2020}
}

Comments

23 pages

R2 v1 2026-06-23T16:05:56.843Z