Monotone functions and quasiconformal mappings on Carnot groups
Let \(G\) be a Carnot group with the homogeneous dimension \(\nu\), \(\Omega\) be a domain in \(G\), \(W^1_p (\Omega)\) and \(L^1_p (\Omega)\), \(1\leq p\leq \infty\), be the Sobolev spaces and the class \(W^1_{p,loc} (\Omega)\) consists of all functions \(f\) such that \(f\in W^1_p (U)\) for every open set \(U\), \(\overline U\subset \Omega\). The author investigates monotone functions of the Sobolev classes and obtains the following results: monotone functions of the class \(L^1_\nu\) are continuous and their family with uniformly bounded Dirichlet integrals is equicontinuous; monotone mappings of class \(W^1_{\nu,loc}\) satisfy Luzin's condition \(({\mathcal N})\). A theorem of structure isomorphism between Sobolev spaces on Carnot groups and a theorem on removable singularities of quasiconformal mappings have been established.
scientific article; zbMATH DE number 1023885
scientific article