JournalsrmiVol. 19, No. 1pp. 143–160

On coincidence of pp-module of a family of curves and pp-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

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 pp-module and the pp-capacity plays an important role. We consider this problem on the Carnot group. The Carnot group G\mathbb{G} is a simply connected nilpotent Lie group equipped with an appropriate family of dilations. Let Ω\Omega be a bounded domain on G\mathbb{G} and K0K_0, K1K_1 be disjoint non-empty compact sets in the closure of Ω\Omega. We consider two quantities, associated with this geometrical structure (K0,K1;Ω)(K_0,K_1;\Omega). Let Mp(Γ(K0,K1;Ω))M_p(\Gamma(K_0,K_1;\Omega)) stand for the pp-module of a family of curves which connect K0K_0 and K1K_1 in Ω\Omega. Denoting by p(K0,K1;Ω)\cap_p(K_0,K_1;\Omega) the pp-capacity of K0K_0 and K1K_1 relatively to Ω\Omega, we show that

Mp(Γ(K0,K1;Ω))=p(K0,K1;Ω)M_p(\Gamma(K_0,K_1;\Omega))=\cap_p(K_0,K_1;\Omega)

.

Cite this article

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

DOI 10.4171/RMI/340