The BV-capacity in metric spaces |
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Authors: | Heikki Hakkarainen Juha Kinnunen |
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Affiliation: | 1. Department of Mathematical Sciences, University of Oulu, P.O. Box 3000, 90014, Oulu, Finland 2. Department of Mathematics, Helsinki University of Technology, P.O. Box 1100, 02015, Helsinki, Finland
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Abstract: | We study basic properties of the BV-capacity and Sobolev capacity of order one in a complete metric space equipped with a doubling measure and supporting a weak Poincaré inequality. In particular, we show that the BV-capacity is a Choquet capacity and the Sobolev 1-capacity is not. However, these quantities are equivalent by two sided estimates and they have the same null sets as the Hausdorff measure of codimension one. The theory of functions of bounded variation plays an essential role in our arguments. The main tool is a modified version of the boxing inequality. |
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