{$p$}-module of vector measures in domains with intrinsic metric on Carnot groups

Abstract

We define the extremal length of horizontal vector measures on a Carnot group and study capacities associated with linear sub-elliptic equations. The coincidence between the definition of the p-module of horizontal vector measure system and two different definitions of the p-capacity is proved. We show the continuity property of a p-module generated by a family of horizontal vector measures. Reciprocal relations between the p-capacity and q-module ( 1/p+ 1/q = 1 ) of horizontal vector measures are obtained. A peculiarity of our approach consists of the study of the above mentioned notions in domains with an intrinsic metric.