A new approach to Sobolev spaces in metric measure spaces

TL;DR

This paper introduces a novel approach to Sobolev spaces in metric measure spaces using a mass transport metric on measures, successfully recovering classical Sobolev spaces in Euclidean settings.

Contribution

The paper develops a new framework for Sobolev spaces in metric measure spaces by defining a mass transport metric on measures and establishing equivalence with classical Sobolev spaces in Euclidean spaces.

Findings

New mass transport metric on measures with compact support

Framework recovers classical Sobolev spaces in Euclidean spaces

Provides a unified approach to Sobolev spaces in metric measure spaces

Abstract

Let $(X,d_X,\mu)$ be a metric measure space where $X$ is locally compact and separable and $\mu$ is a Borel regular measure such that $0 <\mu(B(x,r)) <\infty$ for every ball $B(x,r)$ with center $x \in X$ and radius $r>0$. We define $\mathcal{X}$ to be the set of all positive, finite non-zero regular Borel measures with compact support in $X$ which are dominated by $\mu$, and $\mathcal{M}=\mathcal{X} \cup \{0\}$. By introducing a kind of mass transport metric $d_{\mathcal{M}}$ on this set we provide a new approach to first order Sobolev spaces on metric measure spaces, first by introducing such for real valued functions $F$ on $\mathcal{X}$, and then for real valued functions $f$ on $X$ by identifying them with the unique function $F_f$ on $\mathcal{X}$ defined by the mean-value integral: $$F_f(\eta)= \frac{1}{\|\eta\|} \int f d\eta.$$ In the final section we prove that the approach gives us the classical Sobolev spaces when we are working in open subsets of Euclidean space $\mathbb{R}^n$ with Lebesgue measure.