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 G is a simply connected nilpotent Lie group equipped with an appropriate family of dilations. Let Ω be a bounded domain on G and K0, K1 be disjoint non-empty compact sets in the closure of Ω. We consider two quantities, associated with this geometrical structure (K0, K1; Ω). Let Mp(Γ(K0, K1; Ω)) stand for the p-module of a family of curves which connect K0 and K1 in Ω. Denoting by capp(K0, K1; Ω) the p-capacity of K0 and K1 relatively to Ω, we show that Mp(Γ(K0, K1; Ω)) = capp(K0, K1; Ω). Introduction Let D be a domain (an open, connected set) in R = R ∪ {∞}, and let K0, K1 be disjoint non-empty compact sets in the closure of D. We denote by Mp(Γ(K0,K1;D)) the p-module of a family of curves which connect K0 and K1 in D. Next we use the notation capp(K0,K1; D) for the p-capacity of the condenser (K0,K1; D) relatively to D. The question about coincidence of the p-module of a family of curves and the p-capacity for various geometric configuration has been studied by many authors. For example, in the case when K0 and K1 do not intersect the boundary of D and either K0 or K1 contains the complement to an open n-ball the problem has been solved affirmatively by Ziemer in [23]. Hesse in [10] has generalized this result requiring only (K0 ∪K1) ∩ ∂D = ∅. In the series of papers [2, 3, 4] Caraman has been studying the problem under various conditions on the tangency geometry of the sets K0 and K1 with the boundary of D, D ∈ R. In 1993 Shlyk [16] proved, that the coincidence of the p-module and p-capacity is valid for an arbitrary condenser (K0,K1; D), K0, K1 ∈ D, D ∈ R, (K0 ∪K1)∩∂D 6= ∅. A stratified nilpotent group (of which R is the simplest example) is a Lie group equipped with an appropriate family of dilations. Thus, this group forms a natural habitat for extensions of many of the objects studied in the Euclidean space. The fundamental role of such groups in analysis was envisaged by Stein [17, 18]. There has been since a wide development in the analysis of the so-called stratified nilpotent Lie groups, nowadays, also known as Carnot groups. In the present article we are studying the problem of the coincidence between the p-module of a family of curves 2000 Mathematics Subject Classification. Primary 31B15; Secondary 22E30.