On coincidence of $p$-module of a family of curves and $p$-capacity on the Carnot group
Copyright policy )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) .
- University of Bergen Norway
Analysis on real and complex Lie groups, nilpotent Lie groups, \(p\)-module of a family of curves, p-module of a family of curves, Potentials and capacities on other spaces, \(p\)-capacity, p-capacity, 31B15, Carnot-Carathéodory metrics, 22E30, Potentials and capacities, extremal length and related notions in higher dimensions
Analysis on real and complex Lie groups, nilpotent Lie groups, \(p\)-module of a family of curves, p-module of a family of curves, Potentials and capacities on other spaces, \(p\)-capacity, p-capacity, 31B15, Carnot-Carathéodory metrics, 22E30, Potentials and capacities, extremal length and related notions in higher dimensions
selected citations These citations are derived from selected sources.This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).8 popularity This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.Average influence This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).Average impulse This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.Average
Citations provided by BIP!Popularity provided by BIP!