A relation between classes of functions of bounded mean oscillation on Carnot-Carathéodory spaces and conformal mappings (Q2799766)

The author considers the following general superposition problem:NEWLINENEWLINELet \(F(D)\) and \(F(D')\) be certain normed spaces of functions defined on regions \(D,D'\subset\mathbb{X}\) of a topological space \(\mathbb{X}\) and \(f:D\to D'\) a map. Under which conditions is the operator \(f^\ast:u\in F(D')\to u\circ f\in F(D)\) a linear isomorphism between \(F(D)\) and \(F(D')\) satisfying the inequalityNEWLINENEWLINENEWLINE\[NEWLINEC^{-1} ||u||_{F(D')}\leq||u\circ f||_{F(D)}\leq C||u||_{F(D')}\tag{*}NEWLINE\]NEWLINENEWLINENEWLINEfor an appropriate constant \(C\)? Here, in place of \(\mathbb{X}\) a Carnot-Carathéodory space \(\mathbb{M}\) is considered and \(F(D)\) and \(F(D')\) are the spaces \(\mathrm{BMO}(D)\) and \(\mathrm{BMO}(D')\) of functions of bounded mean oscillation. Using methods of the article [\textit{S. K. Vodop'yanov} and \textit{A. V. Greshnov}, in: Reshetnyak, Yu. G. (ed.) et al., Algebra, geometry, analysis and mathematical physics. Tenth Siberian school, Novosibirsk, Russia, August 14--22, 1996. Novosibirsk: Izdatel'stvo Instituta Matematiki SO RAN. 77--90 (1997; Zbl 0902.43004)] for a homeomorphism \(u\) between two regions of \(\mathbb{M}\), a condition is given in terms of inequalities of type (*) implying that \(u\) is quasi-conformal.NEWLINENEWLINEFor the entire collection see [Zbl 1298.53003].