Tohoku Mathematical Journal
2004
December
SECOND SERIES VOL. 56, NO. 4

Tohoku Math. J.
56 (2004), 553-569

Title $p$-MODULE OF VECTOR MEASURES IN DOMAINS WITH INTRINSIC METRIC ON CARNOT GROUPS

Author Irina Markina

(Received March 28, 2003, revised April 26, 2004)
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.

2000 Mathematics Subject Classification. Primary 31B15; Secondary 22E30, 43A80.
Key words and phrases. Carnot group, extremal length, vector measure, Carnot-Carathéodory metric, capacity.

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