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

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) .