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\(A_ \infty\)-condition for the Jacobian of a quasi-conformal mapping. (English) Zbl 0789.30007

The paper deals with quasiconformal \((qc)\) maps \(f:B^ n \to G\) where \(G=f(B^ n) \subset\mathbb{R}^ n\) and \(B^ n\) is the unit ball of \(\mathbb{R}^ n\). Let us start with the following general remarks. During the past decade many results have been proved that study the following question: When does a local property of a \(qc\) map of \(B^ n\) hold globally (that is, on \(B^ n)\)? The problem becomes trivial if the mapping has a \(qc\) extension to \(\mathbb{R}^ n\) which is the case e.g. if the image domains are “good enough” (e.g. if \(G=B^ n)\). So the goal of the game is to find interesting classes of domains for which this local-to-global passage is still possible. In the present paper the authors study this local-to-global passage problem for integrability of Jacobians and use Gehring’s integrability theorem as a tool. The class of John-domains arises here in a natural way.

MSC:

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
35J30 Higher-order elliptic equations

Keywords:

John-domains
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