A new approach to Sobolev spaces in metric measure spaces. (September 2016)

• Record Type: Journal Article
• Authors: Sjödin, Tomas
• Is Part Of: Nonlinear analysis. Volume 142(2016)
• Page Start: 194
• Page End: 237
• Publication Date: 2016-09
• Subjects: primary 46E35 -- secondary 30L99 31E05; Sobolev space -- Metric measure space -- Mass transport; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2016.04.010
• Languages: English
• ISSNs: 0362-546X

Abstract

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.