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The Sobolev capacity on metric spaces. (English) Zbl 0859.46023

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.

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
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