On coincidence of -module of a family of curves and -capacity on the Carnot group
Irina Markina
University of Bergen, Norway
![On coincidence of $p$-module of a family of curves and $p$-capacity on the Carnot group cover](/_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-rmi-volume-19-issue-1.png&w=3840&q=90)
Abstract
The notion of the extremal length and the module of families of curves has been studied extensively and has given rise to a lot of applications to complex analysis and the potential theory. In particular, the coincidence of the -module and the -capacity plays an important role. We consider this problem on the Carnot group. The Carnot group is a simply connected nilpotent Lie group equipped with an appropriate family of dilations. Let be a bounded domain on and , be disjoint non-empty compact sets in the closure of . We consider two quantities, associated with this geometrical structure . Let stand for the -module of a family of curves which connect and in . Denoting by the -capacity of and relatively to , we show that
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Cite this article
Irina Markina, On coincidence of -module of a family of curves and -capacity on the Carnot group. Rev. Mat. Iberoam. 19 (2003), no. 1, pp. 143–160
DOI 10.4171/RMI/340