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Quasi-conformal continuation of plane homeomorphisms. (English. Russian original) Zbl 0612.30021

Sov. Math. 30, No. 9, 1-5 (1986); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1986, No. 9(292), 3-6 (1986).
A topological embedding \(f: \Sigma\to \bar R^ 2\), where \(\Sigma\) is a set of \(\bar R{}^ 2\), is said to have a bounded module distortion if there exists a constant \(K<\infty\) such that, for each pair of disjoint continua \[ E,F\subset \Sigma,\quad (M(E,F;\Sigma)/K)\leq M[f(E),f(F);f(\Sigma)]\leq KM(E,F;\Sigma), \] where M(E,F;\(\Sigma)\) is the module of the arc family joining E and F in \(\Sigma\) [see e.g. J. Väisälä, Lectures on n-dimensional quasi-conformal mappings (1971; Zbl 0221.30031), 144 p.]. The authors establish criteria of quasiconformal continuation to the whole complex plane \(\bar R{}^ 2\) of a bounded module distortion and, as a consequence, of \(f: D\to \bar R^ 2\), where \(D\subset \bar R^ 2\) is a domain.
Reviewer: P.Caraman

MSC:

30C62 Quasiconformal mappings in the complex plane

Citations:

Zbl 0221.30031
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