Vodop’yanov, S. K.; Ukhlov, A. D. Set functions and their applications in the theory of Lebesgue and Sobolev spaces. II. (Russian, English) Zbl 1091.47027 Sib. Adv. Math. 15, No. 1, 91-125 (2005); translation from Mat. Tr. 7, No. 1, 13-49 (2004). The present paper is a continuation of the authors’ work [Mat. Tr. 6, No. 2, 14-65 (2003; Zbl 1050.47030); English translation in Sib. Adv. Math. 14, No. 4, 78–125 (2004; Zbl 1089.47027)].Let \(D\) and \(D'\) be open sets on a Carnot group \(\mathbb G\) and let \(\nu\) be a homogeneous dimension of \(\mathbb G\). In Section 3.3, the authors define a class of weakly \((p,q)\)-regular mappings and study the question of measure distortion under such mappings for \(1\leq q\leq p\leq\nu\). Namely, a mapping \(\varphi\:D\to D'\) of class \(ACL(D)\) is weakly \((p,q)\)-quasiregular if \(\varphi\) generates a bounded embedding operator \(\varphi^*\:L^1_p(D')\to L^1_q(D)\) by the superposition rule, \(1\leq q\leq p\leq\infty\). The main result is the following Theorem. A weakly \((p,q)\)-quasiregular mapping \(\varphi\:D\to D'\) possesses the \(N^{-1}\) property for \(1\leq q\leq p\leq\nu\).In Section 3.4, it is proven that the operator \(\varphi^*\:L^1_p(D')\to L^1_q(D)\), \(1\leq p<\nu\), is bounded if and only if \(\varphi^*\:W^1_p(D')\to W^1_q(D)\) is bounded. In Section 3.5, the authors study geometric properties of weakly \((p,q)\)-regular mappings for \(\nu<q\leq p\leq\infty\). In Section 3.6, sufficient conditions for a mapping to be weakly \((p,q)\)-regular are given. In Section 3.7, the authors prove an explicit estimate of measure distortion for weakly \((p,q)\)-regular mappings for \(1\leq q\leq p<\nu\). In Section 3.8, it is shown that, under homeomorphisms, the norm of the embedding operator is associated with a countably quasiadditive function defined on open sets in \(D\).In Section 4, the authors consider the problem of finding necessary conditions for extendibility beyond the domain of definition with reduction of the integrability power and the embedding problem in domains with singularities. Reviewer: N. A. Kudryavtseva (Novosibirsk) Cited in 5 Documents MSC: 47B38 Linear operators on function spaces (general) 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations 22E30 Analysis on real and complex Lie groups Keywords:Carnot group; homogeneous space; quasiadditive set function; embedding theorems; \(N\)-property Citations:Zbl 1050.47030; Zbl 1089.47027 PDFBibTeX XMLCite \textit{S. K. Vodop'yanov} and \textit{A. D. Ukhlov}, Mat. Tr. 7, No. 1, 13--49 (2005; Zbl 1091.47027); translation from Mat. Tr. 7, No. 1, 13--49 (2004)