A new 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.