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Asymptotic behavior of the coefficient of quasiconformality and boundary behavior of the mapping of the ball. (Russian) Zbl 0764.30016
For a homeomorphism \(f\) let \(k(f,x)\) denote its linear dilatation at a point of its domain of definition and let \(B^ n\) stand for the unit ball in \(R^ n\). The following theorem, related to earlier work of v.a.Zorich and others, is proved.
Theorem. Let \(f: B^ n\to R^ n\) be a \(C^ 1-\)diffeomorphism such that \[ \int_{B^ n}{k(f,x) dx} < \infty. \] Then \(f\) ha a radial limit at almost every point of the unit sphere.
It is not known whether an analog of his result holds for bounded quasiregular maps in dimensions \(n \geq 3\) or whether even one radial limit exists in this case (cf. Problem (8) on p. 193 of the reviewer’s book “Conformal geometry and quasiregular mappings” (1988; Zbl 0646.30025)).

MSC:
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
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