On coincidence of \(p\)-module of a family of curves and \(p\)-capacity on the Carnot group.
A Carnot group \(G\) is a simply connected nilpotent Lie group, equipped with an appropriate family of dilatations and whose Lie algebra \(G\) splits into the direct sum of vector spaces \(V_1\oplus V_2\oplus\cdots\oplus V_m\). Let \(\Omega\) be a bounded domain of \(G\) and \(K_0\), \(K_1\) be disjoint, compact sets in the closure of \(\Omega\). The author establishes the relation \[ M_p[\Gamma(K_0, K_1;\Omega)]= \text{cap}_p(K_0, K_1;\Omega), \] where \(M_p[\Gamma(K_0, K; \Omega)= \inf_{\rho\in F[\Gamma(K_0, K_1; \Omega)]} \int_\Omega \rho^p\,dx\) is the \(p\)-module \([p\in (1,\infty)]\) of the arc family \(\Gamma(K_0, K_1; \Omega)\) joining \(K_0\) and \(K_1\) in \(\Omega\), \(F[\Gamma(K_0, K_1;\Omega)]\) denotes the set of Borel functions \(\rho: \Omega\to [0,\infty]\) so that, for every locally rectifiable \(\gamma\in\Gamma(K_0, K_1;\Omega)\), \(\sup \int^{l(\gamma')}_0 \rho[\gamma'(t)]\,dt\geq 1\) and the supremum is taken over all arcs \(\gamma'\) such that \(\gamma'= \gamma|_{[\alpha,\beta]}\to \Omega\), \(l(\gamma')\) being the length of \(\gamma\). We precise also that \[ \text{cap}_p(K_0, K_1;\Omega)= \inf_{\varphi\in A(K_0, K_1;\Omega)} \int_\Omega\| X\varphi\|^p\, dx,\quad [p\in (1,\infty)], \] where \(\| X\varphi\|= (\sum^{n_1}_{k=1}| X_{1k}\varphi|^2)^{{1\over 2}}\), \(X\varphi= (X_{11}\varphi,\dots, X_{1n_1}\varphi)\) and \(X_{11},\dots, X_{1n_1}\) is a base of \(V_1\), \(n_1= \dim V_1\). This relation was previously established in the particular case of a domain of \(\overline{\mathbb R^n}= \mathbb R^n\cup \{\infty\}\) [\textit{V. A. Shlyk}, Sib. Math. J. 34, No. 6, 1196--1200 (1993; Zbl 0810.31004); \textit{J. Hesse}, Ark. Mat. 13, 131--144 (1975; Zbl 0302.31009); \textit{W. P. Ziemer}, Mich. Math. J. 16, 43--51 (1969; Zbl 0172.38701) and the reviewer's papers, Ann. Pol. Math. 55, No. 1--3, 37--56 (1991; Zbl 0748.31003); Rev. Roum. Math. Pures Appl. 39, No. 6, 509--553, 555--577 (1994; Zbl 0849.30013 and Zbl 0849.30014) and Bull. Soc. Sci. Lett. Łodz., Sér. Rech. Déform. 45 (20), 79--89 (1995; Zbl 0883.31004)].
scientific article; zbMATH DE number 2005269