Kinnunen, Juha; Martio, Olli The Sobolev capacity on metric spaces. (English) Zbl 0859.46023 Ann. Acad. Sci. Fenn., Math. 21, No. 2, 367-382 (1996). Summary: We develop a capacity theory based on the definition of Sobolev functions on metric spaces with a Borel regular outer measure. Basic properties of capacity, including monotonicity, countable subadditivity and several convergence results, are studied. As an application we prove that each Sobolev function has a quasicontinuous representative. For doubling measures we provide sharp estimates for the capacity of balls. Capacity and Hausdorff measures are related under an additional regularity assumption of the measure. Cited in 71 Documents MSC: 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 28A12 Contents, measures, outer measures, capacities 31B15 Potentials and capacities, extremal length and related notions in higher dimensions 28A80 Fractals 28A78 Hausdorff and packing measures Keywords:capacity theory; Sobolev functions on metric spaces; Borel regular outer measure; monotonicity; countable subadditivity; convergence; quasicontinuous representative; doubling measures; Hausdorff measures PDFBibTeX XMLCite \textit{J. Kinnunen} and \textit{O. Martio}, Ann. Acad. Sci. Fenn., Math. 21, No. 2, 367--382 (1996; Zbl 0859.46023) Full Text: EuDML EMIS