Quasiconformal mappings onto John domains. (English) Zbl 0712.30017

Heinonen, Juha

Rev. Mat. Iberoam. 5, No. 3-4, 97-123 (1989).

Suppose that f is a quasiconformal map of the unit ball \({\mathbb{B}}^ n\) onto a domain D in \({\mathbb{R}}^ n\). Then the first main theorem of this paper provides nine equivalent conditions for D to be a John domain. The conditions involve either the geometry of D or the behavior of f. Similar results in the plane for conformal maps were obtained by Ch. Pommerenke [J. Lond. Math. Soc. 26, 77-88 (1982; Zbl 0464.30012)]; see also R. Näkki and J. Väisälä [Exp. Math. 9, 3-43 (1991)]. The second main theorem extends a subinvariance result of J. Väisälä [Acta Math. 162, No.3, 201-225 (1989; Zbl 0674.30017)], and it describes how a quasiconformal mapping behaves in nice subdomains. This quite general theorem contains as a special case e.g. the fact that if f is a quasiconformal map of a domain D onto the unit ball \({\mathbb{B}}^ n\), then the image of every ball \(B\subset D\) is a uniform domain in \({\mathbb{B}}^ n\); this result was effectively used for plane conformal maps f in establishing the \((1+\epsilon)\)-integrability of \(f'\) on lines [J. L. Fernández, J.Heinonen and O. Martio, J. Anal. Math. 52, 117-132 (1989; Zbl 0677.30012)].

Reviewer: J.Heinonen

MSC:

30C65

Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations

Keywords:

John domains

Citations:

Zbl 0488.30005; Zbl 0464.30012; Zbl 0674.30017; Zbl 0677.30012