A new approach to Sobolev spaces in metric measure spaces

Abstract

Let (X,dX,μ) be a metric measure space where X is locally compact and separable and μ is a Borel regular measure such that 0<μ(B(x,r))<∞ for every ball B(x,r) with center x∈X and radius r>0. We define X to be the set of all positive, finite non-zero regular Borel measures with compact support in X which are dominated by μ, and M=X∪{0}. By introducing a kind of mass transport metric dM on this set we provide a new approach to first order Sobolev spaces on metric measure spaces, first by introducing such for functions F:X→R, and then for functions f:X→[−∞,∞] by identifying them with the unique element Ff:X→R defined by the mean-value integral:Ff(η)=1‖η‖∫fdη. 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 Rn with Lebesgue measure.