Testing the Sobolev property with a single test plan

TL;DR

This paper demonstrates that in certain metric measure spaces, a single test plan suffices to identify the minimal weak upper gradient of Sobolev functions, simplifying the analysis of Sobolev spaces and their properties.

Contribution

It proves that a single test plan can recover the minimal weak upper gradient in a broad class of metric measure spaces, including RCD spaces, improving previous methods.

Findings

A single test plan suffices to recover the minimal weak upper gradient.

On RCD spaces, the test plan can be concentrated on an equi-Lipschitz family of curves.

The approach simplifies the identification of exceptional curves in the weak upper gradient inequality.

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 $p$-modulus. Moreover, on $\sf RCD$ 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.