A new approach to Sobolev spaces in metric measure spaces. (September 2016) - Sjödin, Tomas
Abstract: Let ( X, d X, 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 d M 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 F f : X → R defined by the mean-value integral: F f ( η ) = 1 ‖ η ‖ ∫ f d η . 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 R n with Lebesgue measure.