On coincidence of $p$-module of a family of curves and $p$-capacity on the Carnot group

Irina Markina

doi:10.4171/rmi/340

JournalsrmiVol. 19, No. 1pp. 143–160

On coincidence of $p$ -module of a family of curves and $p$ -capacity on the Carnot group

Irina Markina
University of Bergen, Norway

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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 $p$ -module and the $p$ -capacity plays an important role. We consider this problem on the Carnot group. The Carnot group $G$ is a simply connected nilpotent Lie group equipped with an appropriate family of dilations. Let $Ω$ be a bounded domain on $G$ and $K_{0}$ , $K_{1}$ be disjoint non-empty compact sets in the closure of $Ω$ . We consider two quantities, associated with this geometrical structure $(K_{0}, K_{1}; Ω)$ . Let $M_{p} (Γ (K_{0}, K_{1}; Ω))$ stand for the $p$ -module of a family of curves which connect $K_{0}$ and $K_{1}$ in $Ω$ . Denoting by $\cap_{p} (K_{0}, K_{1}; Ω)$ the $p$ -capacity of $K_{0}$ and $K_{1}$ relatively to $Ω$ , we show that

M_{p} (Γ (K_{0}, K_{1}; Ω)) = \cap_{p} (K_{0}, K_{1}; Ω)

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Cite this article

Irina Markina, On coincidence of $p$ -module of a family of curves and $p$ -capacity on the Carnot group. Rev. Mat. Iberoam. 19 (2003), no. 1, pp. 143–160

DOI 10.4171/RMI/340