Testing the Sobolev property with a single test plan

Abstract

We prove that on an arbitrary metric measure space 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 p-modulus. Moreover, on RCD spaces we can improve our result, showing that the test plan can also be chosen to be concentrated on an equi-Lipschitz family of curves.