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

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